ISNM INTERNATIONAL SERIES OF NUMERICAL MATHEMATICS

INTERNATIONALE SCHRIFTENREIHE ZUR NUMERISCHEN MATHEMATIK

SERIE INTERNATIONALE D'ANALYSE NUMERIQUE

Editors:

eh. Blane, Lausanne; A. Ghizzetti, Roma; A. Ostrowski, Montagnola; J. Todd, Pasadena;

A. van Wijngaarden, Amsterdam

VOL. 21

Spline Functions and

Approximation Theory

Proceedings of the Symposium held at the University of A1berta, Edmonton

May 29 to June 1, 1972

Edited by A. Meir and A. Sharma

1973

Springer Basel AG

ISBN 978-3-0348-5980-6 ISBN 978-3-0348-5979-0 (eBook) DOI 10.1007/978-3-0348-5979-0

© Springer Basel AG 1973

Originally published by Birkhäuser Verlag Basel 1973.

Softcover reprint of the hardcover I st edition 1973

FOREWORV

Around the end of May 1972, a conference on

Approximation Theory was organized at the University

of Alberta, Edmonton. The participants came from all

parts of Canada, the United States and Europe. Since

the majority of talks were related to the theory of

spline functions, we decided to let this volume of the

Proceedings of the conference be entitled "Spline

Functions and Approximation Theory".

We take this opportunity to express our thanks to

all those who participated in the conference or contri­

buted to this volume. Thanks are due to the University

of Alberta and to the National Research Council of

Canada for financial assistance and for the facilities

provided. Professor Ostrowski deserves our special

gratitude for accepting this volume for publication on

behalf of Birkhäuser Verlag. Finally we wish to express

our appreciation to the staff of Birkhäuser Verlag for

their courtesy and valuable co-operation.

LIST OF PARTICIPANTS

C. DeBoor (Lafayette)

R. Bojanic (Columbuc)

Q. Rahman (Montreal)

I.J. Schoenberg (Madison)

W.A. Al-Salam (Edmonton)

A.M. Ostrowski (Basel)

M.Z. Nashed (Madison)

J.M. Munteanu (Baitimore)

L.L. Schumaker (Austin)

E.W. Cheney (Austin)

J. Fields (Edmonton)

A. Meir (Edmonton)

D.W. Boyd (Vancouver)

M. Ismail (Edmonton)

D. Leeming (Victoria)

M. Marsden (Pittsburgh)

E. Schmidt (Calgary)

S.W. Jackson (Edmonton)

S. Riemenschneider (Edmonton)

B. Cairns (Edmonton)

R. Gopalan (Edmonton)

J.C. Fung

P. Kumar

H. Berens (Santa Barbara)

Z. Ditzian (Edmonton)

G.G. Lorentz (Austin)

T.N.E. Greville (Madison)

C. Davis (Toronto)

R.S. Varga (Parma)

J.W. Jerome (Evanston)

E.G. Straus (Los Angeles)

F. Richards (Edmonton)

R. DeVore (Edmonton)

P.M. Gauthier (Montreal)

A. Sharma (Edmonton)

A. Al-Hussaini (Edmonton)

A. Giroux (Montreal)

R. Gervais (Montreal)

R. Pierre (Montreal)

G. Votruba (Missoula)

C. Nasim (Calgary)

S. Cabay (Edmonton)

C.P. May (Edmonton)

R. Gaudet (Edmonton)

D.S. Goel

W.Y. Chum

CONTENTS

Berens, H.: Pointwise Saturation. • • 11

Davis, C.: A Combinatorial Problem In Best Uniform Approximation. • • • • • • • • • • 31

DeBoor, C.: Good Approximation By Splines With Variable Knots • • •• •.•• 57

DeVore, R. & Richards, F.: Saturation And Inverse Theorems For Sp1ine Approximation. • • 73

Ditzian, Z. & May, C.P.: Saturation Classes For Exponential Formulae Of Semi-Groups Of Operators 83

FieZds, J.L. & IsmaiZ, M.E.: On Some Conjeetures Of Askey Coneerning Completely Monotonie Funetions. 101

Gauthier, P.M.: Une Applieation De La Theorie De L'Approximation A L'Etude Des Fonetions Holomorphes 113

Jerome, J.W.: Linearization In Certain Noneonvex Minimization Problems And Generalized Sp1ine Projeetions. • . . • . . • • . . . 119

Lyche, T. & Schumaker, L.L.: On The Convergenee Of Cubie Interpolating Splines • 169

Motzkin, T.S., Sharma, A. & Straus, E.G.: Averaging Interpolation. • • • . • • • • • 191

MUnteanu, M.: On The Construetion Of Multidimen-sional Splines . • • • • • • . • • . . • • 235

Ostrowski, A.M.: On Error Estimates APosteriori In Iterative Proeedures. • • . • • • • • • • • • •• 267

Schoenberg, I.J.: Sp1ines And Histograms. . • •• 277

DeBoor, C.: Appendix To 'Sp1ines And Histograms' By I.J. Sehoenberg • •• •• • . . • • • • •• 329

Straus, E.G.: Real Ana1ytie Funetions As Ratios Of Abso1ute1y MOnotonie Funetions. • • • • • • •• 359

ABSTRACTS

DeVore, R.: Inverse Theorems For Approximation By Positive Linear Operators. • • • • • • • • • • 371

Meir, A. & Sh~a, A. : Laeunary Interpolation By Sp1ines. • • • • • 377

Morr-is, P.D. & Cheney, E.W.: Stabi1ity Properties Of Trigonometrie Interpolation Operators . . . •. 381

Vapga, R.S.: Chebyshev Semi-Diserete Approximation For Linear Parabo1ie Problems. • • • • • • • • •• 383

POINTWISE SATURATION

Hubert Berens

O. The phenomenon of saturation which is associated

with many approximation processes can be considered as

a form of a differentiation process, and the questions

asked about saturation are ana10gues of questions

asked about differentiation.

To make the claim more precise, let us look at an

examp1e instead of giving formal notations and

definitions. The examp1e to be considered are the

Bernstein polynomials.

1. Let e[O,l] denote the space of all rea1-va1ued,

continuous functions on [0,1] under the supremum

For f E e[O,l] the th Bernstein norm. an , n

po1ynomia1 B f(x), n = 1,2, ••• , is defined as n

11

12

where

B f(x) = n

n k I f(-)Pk (x) k n ,n

=0

H. Berens

S.N. Bernstein introduced these polynomials in 1912/13

and proved

1im B f(x) = f(x) on [0,1], n n-+oo

uniform1y in x for each f E C[O,l] •

In 1932 E.V. Voronovskaya estab1ished the fo110w­

ing asymptotic relation

1im n{B f(x) - f(x)} n n-+oo

= x(l-x) f"(x) , 2

whenever the right-hand side exists.

This relation shows that the order of approxima­

tion of a non-linear twice differentiab1e function f

by B f n

is bounded by O(l/n) independent of its

actua1 order of smoothness, and one says with

J. Favard that the approximation process towards the

identity I on C[O,l] given by the Bernstein

Pointwise Saturation 13

operators {B n

n > 1} is saturated with order O(l/n)

as n -r 00 •

On the other hand, we may consider the sequence of

operators {n[B -I] : n > 1} n -

on C[O,l] as a process

converging to the differential operator D as n -r 00 ,

where

Df(x) = x(l-x) f"(x) . 2

From this point of view it is natural to ask whether

Voronovskaya's relation is invertib1e. First, does

1im {B f(x) - f(x)} 0 n n-roo

pointwise on (a,b) c [0,1] for some f E C[O,l]

imp1y that f"(x) exists on (a,b) and equa1s zero,

i.e., f is linear on [a,b]?

This was conjectured by G.G. Lorentz in his mono­

graph [14] on Bernstein polynomials in 1953 and proved

by B. Bajsanski-R-Bojanic in a note [3] in the BAMS

in 1964. 1t is a consequence of the fo11owing

THEOREM A. Let f E C[O,l] be such that for aZZ

x E (a,b) c [0,1]

14 H. Berens

° < lim n{B fex) - fex)} , - n n-?<X>

then f is convex on [a,b].

The result above is known as the pointwise 0-

theorem for the Bernstein polynomials, and it is now

meaningfu1 to ask the fo110wing more general question:

Let f E e[O,l] , and let g be a function on

(a,b) c [0,1] • Does

1im n{B fex) - fex)} n n-?<X>

= x(l-x) g(x) 2

on (a,b)

imply that f is twice differentiab1e on (a,b) and

f"(x) = g(x) ?

The answer is affirmative in the fo110wing sense:

THEOREM B. Let f E e[O,l], g E L(a,b) , finiteZy

vaZued, where (a,b) C [0,1] . If

1im n{B fex) - fex)} < x(l-x) g(x) < 1im n{B fex) - fex)}

-- n 2 - n n-?<X> n-?<X>

on (a,b), then

x t

fex) = Ax + B + f dt f g(u)du a a

Pointwise Saturation 15

on [a, b ]. A and B are two appropriate constants.

This is a pointwise saturation theorem and it

generalizes the already elassieal saturation theorem due

to G.G. Lorentz in 1963 [15, p. 104].

THEOREM C. Let f E C[O,l] , and let M be a positive

constant such that

nIBnf(x) - f(x)1 ~ x(~-x) M + 0x(l) on (a,b) c [O,lJ ,

then f is continuously differentiable on [a,b] and

I f' (x) - f' (x) I ~ M I x - xl , x, X E [a, b] ,

and vice versa.

Theorems A and Bare elosely related to a result of

H.A. Schwarz on the eharaeterizations of eonvex funetions

through seeond symmetrie derivatives and C. de la Vallee­

Poussin's extension, a fundamental lemma in the theory of

trigonometrie series. The lemma reads:

Let f E C[a,b] ,let g E L(a,b) , finitely­

valued, and let

f(x+t) + f(x-t) =

2 t" ° . If

16 H. Berens

1im ~ {Ltf(x) - fex)} t~O+ t

on (a,b) , then

x t fex) = Ax + B + f dt f g(u)du on [a,b].

a a

The proof is done by reducing it to the case

g(x) = 0 , which is essentia1ly Schwarz resu1t, and this

is obtained by a theorem in Lebesgue theory on majorant

and minorant functions due to de la Vallee-Poussin, see

G.H. Hardy-W.W. Rogosinski [10, p. 90].

Hence one may interpret the three theorems for the

Bernstein polynomials as analogues of the classical

results of Schwarz and de la Vallee-Poussin on second

symmetrie derivatives. These theorems can further be

extended to a whole class of approximation methods which

satisfy a Voronovskaya condition.

2. Let v,w be functions in C[a,b] , strictly positive

on (a,b) , and let

Pointwise Saturation

x x ~(x) = J v(t)dt I/I(x) = J w(t)dt

a a

x and ~(x) = J I/I(t)v(t)dt

a

For a function f in C[a,b] , define the differential

operation Df(x) at a point x E (a,b) by

1 {f' (x)}' Df(x) = DI/ID~f(x) = w(x) v(x) ,

17

whenever the right-hand side is meaningfu1, see S. Kar1in­

W. Studden [13, eh. XI].

Let {L n > 1} be a sequence of positive linear n

transformations on C[a,b] into itse1f, let O. :n~l} n

be a sequence of positive rea1s tending to ~ as n ~ ~ ,

and let p(x) be a function in C[a,b] , strict1y positive

on (a, b) • We say {L : n > 1} n

satisfies a

Voronovskaya condition if

1im Ä {L fex) - fex)} = p(x)Df(x) , x E (a,b) , n~ n n

whenever Df(x) exists.

Under these conditions

18

1im L f(x) = f(x) n n-+<x>

(f e: C[a,b])

pointwise for each x e: (a,b) • We have

THEOREM AI. FoX' a funation f e: C[a,b]

o < 1im A {L f(x) - f(x)} on (a,b) - n n n-+<x>

H. Berens

if, and on"ly if, f is aonvex with X'espeat to {1,~} on

[a,b] i.e.,

{~(x1)-~(x )}f(x) < {~(x)-~(x )}f(x1) + {~(x1)-~(x)}f(x ) 0-0 0

x < x < xl • 0- -

COROLLARY. Let f e: Cla,b] be suah that

1im A {L f(x) - f(x)} = 0 on (a,b) , n n n-+<x>

then f is "lineax> with X'espeat to {1,~} , i.e.,

A + B~(x) , wheX'e A and B ax>e aonstants.

THEOREM BI. Let f e: C[a,b] , and "let g e: L(a,b)

finite"ly va"lued. If foX' eaah x e: (a,b)

f(x)

be

1im A {L f(x)-f(x)} < p(x)g(x) < 1im A {L f(x)-f(x)} , --nn - - nn n-+<x> n-+<x>

=

Pointwise Saturation 19

then

x t

fex) = A + B~(x) + J v(t)dt J g(u)w(u)du on [a,b] , a a

whepe A and B aPe aonstants.

These theorems are best possib1e in the fo1lowing

sense: If the limit relations are violated at even one

point in (a,b) then the conclusion does not hold.

THEOREM C'. Let f € C[a,b] ~ and Zet M be a positive

aonstant.

A IL fex) - f(x)1 < p(x)M + 0 (1) as n + ~ n n - x

if~ and onZy if~ D~f exists and beZongs to C[a,b] and

Theorem A' is independently due to several authors.

G. Mühlbach [17] used divided differences and results of

T. Popoviciu on generalized convexity, extending results

of V.A. Amel'kovic [1], in 1966. The proofs given by

J. Karamata-M. Vuilleumier [12] and G.G. Lorentz-

L.L. Schumaker [16] use in principal the arguments in the

proof of Schwarz' lemma, mentioned above. Theorem C' is

in the above form due to G.G. Lorentz-L.L. Schumaker, loc.

20 H. Berens

cit., who extended the versions of V.A. Amel'kovi~ and

G. Mühlbach. Theorem B' is due to the author r6].

The conditions imposed on the sequence of operators

{L : n > I} are satisfied for a large class of appro-n

ximation processes. In addition to the Bernstein opera­

tors, we want to mention the Bernstein power series

introduced by W. Meyer-Konig and K. Zeller, the extension

due to E.W. Cheney and A. Sharma, the Szasz operators,

the Gauss-Weierstrass operators, etc.

A sequence of operators which does not belong to

this class are the Fejer-Hermite operators: Let

f E Cr-l,l] ,

T2 (x) n

H f (x) = --..:..:..,::--n 2

where T (x) n

is the

x. = cos(2i-l)~/2n • 1.

n

th n

n L f(x)

i=l

I-xx. 1.

Chebyshev polynomial and

{H : n > I} forms a sequence of positive linear n

interpolation operators which does not satisfy a

Voronovskaya condition, at least not pointwise.

Pointwise Saturation 21

3. The phenomenon of saturation is best studied for

summability methods of Fourier series and integrals, see

the monograph [7] of P.L. Butzer-R.J. Nessel or the

lecture notes [8] of R. DeVore.

Let fex) be a real-valued, 2~-periodic integrable

function on the real line. lts Fourier series is denoted

by

a ()O

fex) ~ ~ + I ~ (x) 2 k=l-K.

where and are its Fourier coefficients.

conjugate series of f is defined by

()O

Hf(x) ~ I Bk(x) k=l

The

Concerning pointwise saturation, a first result was given

by V.A. Andrienko [2] in 1968 for the Fej~r means. The

th n Fejer means of the F.s. of a function f, defined by

a 0" (f;x) = ~ + n 2

n k I (1 - n+l)~(x)

k=l n=1,2, ••• ,

form a positive summation methode Andrienko proved

22 H. Berens

THEOREM D. Let f e: L21T be finiteZy-vaZued and suah that

0n(f;x) aonve~es to fex) as n ~ ~ fop aZZ x in

some intepvaZ (a,b). If

(1) lim (n+l){o (f;x) - fex)} = 0 n n-+oo

fop aZZ x e: (a,b) ~ then~ fop aZmost aZZ x~ Hf(x) is

a aonstant funation on (a,b). Mopeovep~ if f e: C21T

then the aonaZusion pemains tpue even if (1) is vioZated

in a denumepabZe set of points.

To establish the connection to the previous sections,

let X21T be one of the spaces L~1T' 1 2 p < ~ , or C21T

endowed with the usual norm. The following asymptotic

relation is weIl known:

If fand [Hf]' belong to X21T , then

(2) lim (n+l){on(f) - f} = -[Hf]' in X21T-norm. n-+oo

The differential operator (d/dx)H on the right-hand

2 2 side of (2) and the operator d /dx are connected

through

Pointwise Saturation

d -H = dx

i.e., here we are dea1ing with an asymptotic relation

i d f i 1 f _d 2/dx2 , converg ng towar s a ract ona power 0

see [7].

23

The asymptotic relation (2) imp1ies the saturation

theorem for the Fejer means which is due to G. A1exits

for C2~' in 1941, and G. Sunouchi-C. Watari in general,

in 1958/59.

The pointwise saturation theorem for the Fejer means

reads:

THEOREM E. Let f € L2~ be such that

finitely for all X in some interval

1im ° (f;x) = fex) n

n~

(a,b) " and let

g € L(a,b) be finitely-valued and satisfy

(3) 1im (n+1){on+1(f;x) - fex)} = g(x) n~

00

pointwise on (a,b). Then ~1~(x)/k is the P.s. of

some F € L2n and for almost all x € (a,b)

x t

F(x) Ax + B + f dt f g(u)du a a

24 H. Berens

The proof of these two theorems rests on the

following observation: Let

(4) 00

l (-k)~(x) k=l

be the trigonometrie series assoeiated with [Hf]' • The

limit in (3) exists finitely if, and only if,

n k lim cr ([Hf]',x) = lim l (1 - n+l)(-k)~(x) n~ n n~ k=l

exists, and both limits are equal. This reduees the

theorem to a uniqueness problem of a Fejer summable

trigonometrie series. The uniqueness theorem ean be

obtained either from S. Verblunsky's uniqueness theorem

for Abel summable trigonometrie series (cf. A. Zygmund

[21, p. 352ff]), or from results due to F. Wolf [20]

about (C,A)-summable series.

As a eonsequenee of Theorem E we have the following

COROLLARY. Let f € C2~' g € L2~ , fineteZy-vaZued. If

(3) hoZdS tpue fop aZZ x e~aept possibZy on a denumep­

abZe set, then Hf is absoZuteZy aontinuous and

[Hf]'(x) = -g(x) a.e.

Pointwise Saturation 25

The coro11ary substantia11y weakens the asymptotic

relation (2) for continuous functions.

The resu1ts stated for the Fej~r means can be ex­

tended to the typica1 means of order y: For an

f E L21T

n y [T f](x) = L (1 - k )~ (x)

n,y k=l (n+1)y -K (y > 0, n = 1 2 ) , ,... .

In this case,

n~

1im (n+1)y{T f - f} n,y

See [4] and G. Sunouchi [19]; see also f5] for a point­

wise saturation theorem for the Abe1 means.

R. DeVore [9] proved a general pointwise o-theorem

for approximation methods on C21T which commute with

translations and which have the saturation phenomenon.

To be precise, let f E C21T and let

a 00

~ ~ + L P ~ (x) ,. n = 1,2,... , 2 k=l k,n-K

26 H. Berens

where Iln is a positive even Borel measure on [-lT,lT] lT

with Pk = (l/lT) f cos k.x dll (x), Po,n = 1 , and such ,n n -lT that

l-p lim PI n = 1 and lim k,n = 1/Ik > 0 l-p n-+oo ' n-+oo l,n

for each k = 1,2, .••.

The last conditions assure that {L : n ~ I} is n

an approximation process saturated with order O(l-P l ) ,n as n -+ 00 •

THEOREM F. Let f be in C2lT suah that fo1' aZZ

IL fex) - fex) I = 0 (1 - P ) as n -+ 00 , n x l,n

then f is a aonstant-vaZued funation.

R. DeVore's proof is indirect and based on an

analysis of the support of Iln in [-lT,lT] as n-+ oo •

The associated pointwise saturation theorem is not known,

and it is unlikely to hold true in the given generality,

(see [8]). However, it seems to be of interest to find

sufficient conditions which guarantee a pointwise theorem

for classes of approximation processes. A first step in

this direction has been done by T. Hedberg [11], a young

Pointwise Saturation

Swedish mathematieian, and this even for Fourier series

and integrals in severa1 variables.

5. Riemann's theory of trigonometrie series and

integrals in m-dimensions was 1arge1y deve10ped by

27

V.L. Shapiro, (see [18]). Hedberg studied uniqueness

theorems for summab1e trigonometrie series and integrals,

where the summation method is given as a eonvo1ution

integral.

As an app1ieation he proved, e.g., the fo11owing

pointwise saturation theorem:

THEOREM G. Suppose that K is a positive radiaZ kerneZ

on Em whiah satisfies (iJ J Kdx = 1 and (iiJ

Em R + 00 , and set J Kdx = o(R-2)

Ixl~R as ~(x) = R~(Rx) •

If f is a bounded aontinuous funation in Em for whiah

2 1im R {f*~(x) - fex)} = g(x) , R+oo

at eaah point x E ~ , where g is finite and ZoaaZZy

integrabZe, then ~f = g •

~ is the Lap1aee operator. Hedberg also proves a

28 H. Berens

pointwise o-theorem for the Poisson integral on Em and

Tm , and gives an independent proof of the pointwise

saturation theorem for the Abe1 means of Fourier series

and integrals on the real 1ine.

REFERENCES

1. Ame1'kovic, V.G.: A theorem converse to a theorem of Voronovskaya type, Teor. Funkei!, Funkciona1 Anal. i Pri1ozen, Vyp 2 (1966), 67-74.

2. Andrienko, V.A.: Approximation of functions by Fejer means, Siberian Math. J. 9 (1968), 1-8.

3. Bajsanski, B. and Bojanic, R.: A note on approxi­mation by Bernstein polynomials, Bu11. Amer. Math. Soc. 70 (1964), 675-677.

4. Berens, H.: On pointwise approximation of Fourier series by typica1 means, T6hoku Math. J. 23 (1971), 147-153.

5. Berens, H.: On the approximation of Fourier series by Abe1 means, J. Approximation Theory (accepted for pub1ication).

6. Berens, H.: Pointwise saturation of positive operators, J. Approximation Theory 5 (1972), 135-146.

Pointwise Saturation

7. Butzer, P.L. and Nessel, R.J.: Fourier Analysis and Approximation, Basel 1970.

29

8. DeVore, R.: Approximation of continuous functions by positive linear operators, Lecture Notes 1970.

9. DeVore, R.: A pointwise "0" saturation theorem for positive convo1ution operators, Proceedings of the Conference on Linear OperatoIS and Approximation, Oberwolfach 1971.

10. Hardy, G.H. and Rogosinski, W.W.: Fourier Series, Cambridge 1944.

11. Hedberg, T.: On the uniqueness of summab1e trigonometrie series and intergrals, Ark. Mat. 9 (1971), 223-241.

12. Karamata, J. and Vui11eumier, M.: On the degree of approximation of continuous functions by positive linear operators. Mathematics Research Center, U.S. Army, Madison, Wisconsin.

13. Kar1in, S. and Studden, W.: Tchebycheff Systems. New York 1966.

14. Lorentz, G.G.: Bernstein Polynomials. Toronto 1953.

15. Lorentz. G.G.: Approximation of Functions. New York 1968.

16. Lorentz, G.G. and Schumaker, L.L.: Saturation of positive operators, J. Approximation Theory 5 (1972), in printe

17. Mühlbach, G.: Operatoren vom Bersteinsehen Typ, J. Approximation Theory 3 (1970), 274-292.

30 H. Berens

18. Shapiro, V.L.: Fourier Series in Several Variables, Bull. Amer. Math. Soe. 70 (1964), 48-93.

19. Sunouehi, G.: Pointwise approximation of funetions by typieal means of Fourier series (to be pub1ished).

20. Wolf, F.: On summable trigonometrie series: an extension of uniqueness theorems, Proe. Land. Math. Soe. (2) 45 (1939), 328-356.

21. Zygmund, A.: Trigonometrie Series. Vol. I, Cambridge 1959.

A COMBINATORIAL PROBLEM IN BEST UNIFORM APPROXIMATION

Chandler Davis

PROBLEM: Given a funation f of one vaPiable, to

minimize

Ilf - gll= sup If(t) - g(t)1 t

among all g whiah aPe monotonia on at most n sub­

intervals.

1. ROW TO POSE TRE PROBLEM AND WHY

31

Sometimes all we really want to know about a

function is the sequence of its increases and decreases.

When it is given empirically or by an imperfect formula,

the imperfection we most regret may be the occurrence of

small spurious ups and downs of the graph. Indeed some

of them may not be so small that we feel safe and easy

about modifying the function so as to flatten them out.

32 C. Davis

Let us put the matter quantitatively.

DEFINITION~ A function g defined on areal interval

[a,b] will be said to have 'oscillation order' n in

case there exists a partition [a,b] = [to,t1 ] U [t1 ,t2]

U ••• U [t l,t] such that the restrictions of g to n- n

[t. l,t.] (j = l, ••. ,n) are, in alternation, non-con-J- J

stant increasing and non-constant decreasing functions.

Por a function defined on a subset (in particular, a

discrete subset) of an interval, the terminology is the

scune, wi th the taci t understanding that a symbo l for an

interval denotes the intersection of that interval wi th

the domain.

Thus there are two kinds of functions of oscil­

lation order n : those which begin on the first sub­

interval with an increase, and those which begin with a

decrease; but if any of the n restrictions were con­

stant, we would be obliged to change the partition to

one with fewer subintervals and say the oscillation

order was properly less than n. Note that the

definition does cover the extreme situation in which the

domain of g is just {t , t 1 , •.• , t } . o n

Assume the given function f has some finite

oscillation order N. (No interesting cases are there-

A Combinatoria1 Problem in Best Approximation

by exc1uded.) We ask whether any g with osci11ation

order ~ n is reasonab1y c10se to f But by what

metric shou1d we judge what is "c1ose"?

One reasonab1e definition wou1d be by an LP(~)

norm, for some measure ~ on the domain, and

1 ~ p < ~ For p = 2, n = 1 this is c10se to a

33

problem which has been studied by statisticians [1]. A

second possibi1ity, which appears the most natural in

case the functions in question are frequency functions,

is the Kantorovic-Rubinstein metric [3]. In this metric

the distance from f to g is the solution of a trans­

portation problem: to move a unit mass distributed

according to f unti1 it is distributed according to

g , at minimum total cost (mass times distance). Here

I fo11ow a third alternative definition, the supremum

norm. It is especia11y suited to those situations which

are unaffeated by order-ppeserving ahanges of the inde­

pendent variabLe. Thus this work is descended, in

spirit though not in detail, from the qualitative theory

of Cebysev systems, especia11y [4], [5], [2].

By assumption, there are N subinterva1s,

exhausting [a,b] , on each of which f is monotonie,

with alternation between increase and decrease on

successive interva1s. Let [te 1,t.] be one of the J- J

34 C. Davis

subintervals, whose endpoints do lie in the domain, and

let g be any proposed approximant to f. Now suppose

g to be replaced by that function g' which has the

same values as g except that at points of the domain

lying in ]t. l,t.[ J- J

(if any) it is redefined so that

g' is, on [t. l,t.] , a linear transform of f J- J

The

change can not increase the distance away from f, nor

can it increase the oscillation order of the approximant.

Accordingly there was no need to consider the irrelevant

added structure of g in the first place; that is to

say, the whole problem under study is determined by the

sequence of values of f at the points of its domain

where it reverses the sense of monotonicity. Since the

parametrization is also without effect, we are free to

declare these points to be equally spaced. Summing up,

there is no loss in generality in confining ourselves to

the following discrete version:

PROBLEM: Given a function f on {O,l, •.. ,N} , such

that the sequence (f(j) - f(j-l)~=l is alternating

in sign (strictly)~ to minimize

max If(j) - g(j)1 j

among all functions g on {O,l, ... ,N} which are of

oscillation order at most n (n < N) •

A Combinatorial Problem in Best Approximation

These assumptions and notations will be retained

throughout the rest of the paper.

35

The main features of the problem appear already in

the simplest ease.

Example 1; N = 2, n

j

f(j)

1

o

o

1

3

Define f by

2

1

We are to find its best approximation by a monotonie

funetion (funetion of oseillation order < 1). It is

easy to see that the best approximating g must have

gel) = g(2) = 2 ; but there is no uniqueness, g(O) ean

be anything subjeet to -1 ~ g(O) < 1 .

Apparently the main objeetive should be an effieient

algorithm leading to some best approximant. I believe

this is aehieved satisfaetorily by the analysis in

Seetion 3,4. First I state in Seetion 2 some of the

neater results of the analysis, not beeause they are its

whole aim, but in order to show at onee how the be­

haviour observed in Example 1 generalizes.

36 C. Davis

2. SOME OF THE RESULTS

DEFINITION:. A ' subfunction' of f is the restriction

of f to a subset {i ,il, ..• ,i} of {O,l, ••. ,N} o m

(m ~ N; i k_l < i k) ~ such that the sequence

(f(ik) - f(ik_l»~=l is alternating in sign (strictly).

(The subset deterrnines the subfunction~ and it wiU be

harmless to say the subset is the subfunction~ f being

fixed.) The ' discY'epancy' of the subfunction

{io ,il ,··· ,im} is mink !f(ik) - f(ik_l )! •

THEOREM: Let the maximal discY'epancy~ among aU sub­

functions of osciUation oY'deY' exactZy n+ 1 ~ be 28.

Then the minimum of 11 f - g 11 ~ among aU g of

oscillation oY'deY' ~ n ~ is 8.

Half of this can be proved at once. Let

{io,il , •.. ,in+1} be a subfunction of discrepancy 20 ,

and let g be any function with IIf - gll = 8' < 8 •

Then because each

each g(ik) - g(ik_1) has to be non-zero and of the

same sign as f(ik) - f(ik_1). Therefore g has

A Combinatorial Problem in Best Approximation 37

oscillation order > n + 1 •

The inequality in the other direction will be

proved along with the construction of certain special

best approximants. Part of the details will be stated

in this Section and proved in Section 3.

DEFINITION: The subfunction {i ,il, ... ,i} will be o m

called 'critical' in case (for each k 0,1, ... ,mJ

f(i ) = m~x {f(·) k ml.m J

the signs "max" and "min" being chosen in alternation as

k increases. Here the otherwise undefined symbol i_I

is to be interpreted as o ~ whether or not i = 0 ; o

similarly~ i mtl is to be interpreted as N.

Convention: To say a subfunction {io,il , .•• } 'has

maximal discrepancy' implies not only that

mink /f(ik ) - f(ik_l )/ has the largest possible value,

but also that this minimum is attained for the smallest

possible number of different k.

PROPOSITION: Among those subfunctions of order m

having maximal discrepancy~ at least one is critical.

38 C. Davis

THEOREM: Let {io,il, ••• ,in+l } be a critiaaZ sub­

funation having maximaZ disapepanay, and Zet

mi~ If(ik) - f(ik_1)I be attained fop k = k' • Then

thepe is a funation g minimizing 11 f - g 11 among

funations of osaiZZation oPdeP < n and having the =

fupthep pPOpepty that g(ik) = f(ik) fop aZZ

k ~ k'-l,k' ; exaept that k = 0 must aZso be exaZuded

if i ~ 0 , and k = n + 1 must aZso be exaZuded o

if i n+l ~ N •

ExampZe 2: N = S •

j 0 1 2 3 4 S

f(j) 0 3 2 S 0 2

Then the subfunction {0,1,4,S} has maximal discrepancy

but is not critical. Similarly for {2,3,4,S} •

3. FROM CRITICAL SUBFUNCTION TO BEST APPROXlMANT

The first aim is to prove the Proposition stated

in Section 2.

If there is some k' (0 < k' < m) such that = = f(ik ,) is greater than f(ik'_l) or f(ik'+l) but

A Combinatorial Problem in Best Approximation

max {f(j)

then redefine i k , to equal a j where that maximum

is assumed; similarly for minima. Each such change

increases either one or two of the If(ik - f(ik_l )I

39

and leaves the others unchanged, so their minimum, the

discrepancy, is if anything increased. For the same

reason we note that, in any step which does not alter

that minimum, the number of different k at which it is

assumed is if anything decreased. After a finite number

of iterations the process can not be continued, and the

subfunction is then critical.

Nowassume {io,il, .•. ,in+l } is a critical sub­

function having maximal discrepancy 28. I will con-

struct a function g of oscillation order m < n such =

that Ilf - gll = 8

First I will choose a set of points {h ,hl, •.• ,h } o m

from among {O,l, ••. ,N} (hk_ l < hk). They will serve

as guides in the construction of g; hl, .•• ,hm_l are

to be local extrema of g , while

mere navigational aids.

h o

and h m

Fix k' such that If(ik ,) - f(ik'_l)1 = 28

are

40 C. Davis

(1 ~ k' ~ n + 1) •

Gase A: k' = 1. In this case {il ,i2 , ••• ,in+l } will

be chosen as the hk , and the function g we get will

have oscillation order n.

The symmetrical case with k' = n + 1 is given

corresponding treatment and will also be called Case A.

Gase B: j < i with o

f(j) < f(io) - 28. Let jo be some j which minimizes

f (j) among all j < i o

In this case, the hk will

be {jo,io,il , .•• ,in+l}\{ik'_l,ik } , and we will again

get a function g of oscillation order n.

There are symmetrical variants of this case too.

(Apparently a subfunction could fall under Case A

as regards the left end of the interval and also fall

under Case B as regards the right end. Such an ambiguity

is actually prohibited by the hypotheses, but we won't

have need of that fact. If a subfunction is in both

Case A and B as regards the same end, then folIoweither

rule, they will give the same function.)

Gase G: We are not in Case A or B. In this case,

A Combinatoria1 Problem in Best Approximation

{io, ..• ,ik'_2,ik'+1, ... ,in+1} will be chosen as the

hk ' so that we will get a function g of oscia11ation

order n - 1 •

In the rest of the Section, let {h ,h1 , •.• ,h } o m

(m = n or n - 1) be a sequence obtained by one of

the ru1es A, B, or C, and take without 10ss of

genera1ity f(ho) < f(h1) •

Construation of the approximant: Define g(hk) to be

f(hk) for 1 < k < m - 1. For j' E ]h1 ,h2 [ , define

2g (j ') = min {f (j) : h1 < j < j'}

+ max {f(j) j' < j < h 2 }

Simi1ar1y for Jh2,h3[, •.• ,Jhm_2,hm_1[. For

j' E [O,h1 [ , define

2g(j') = max {f(j) : 0 ~ j ~ j'}

+ min {f(j) ., < . < h } J = J 1

Simi1ar1y for ]hm_1 ,N] •

There are severa1 verifications to make.

41

42 C. Davis

First, how many intervals of monotonicity does this

g have? It is (non-strictly) increasing and (non­

strictly) decreasing, alternately, in the intervals

[O,hl],[hl,h2], ••• ,[hm_2,hm_l]' [hm_l,N]. To see

this requires two observations: (i) g is non-increasing

on (say) ]hl ,h2 [ because each term in the above

definition is non-increasing, and (ii) f(hl ) ~ f(j)

for hl < j ~ h2 by a short argument using the

derivation of {ho,hl , .•. } from a critical subfunction;

similarly for the other subintervals. Furthermore, on

each of these m intervals the value of g changes by

at least 0 (as you may check from the definitions) and

so is surely not constant. Therefore g really does

have oscillation order m.

Next, the value of Ilf - gll is really < o. It

is convenient to prove this in terms of an auxiliary

notion.

DEFINITION: On one of the intervaZs [O,hl ],[h2 ,h3],

[h4 ,hS]' •.. where g is inareasing, the 'retrogression'

of f is defined to be ° if f is aZso (non-striatZy)

inareasing there; othe~ise, it is defined to be

max {f (j ') - f (j ") } taken over pairs j' < j" in the

intervaZ. SimiZarZy, on one of the intervaZs [hl ,h2],

A Combinatoria1 Problem in Best Approximation 43

[h3,h4], .•. where g is deareasing, the 'retrogression'

of f is defined to be 0 if f is aZso (non-striatZy)

deareasing there; otherwise, it is defined to be

max {f(jlt) - f(j')} taken over pairs j' < jlt in the

intervaZ.

It is easy to see that the gwhich has been de­

fined will be within 8 of f if and on1y if f has

retrogression ~ 28 on each of the m interva1s.

According1y we want to estab1ish the 1atter fact.

To this end, consider first an interval of mono­

tonicity obtained (by any of Cases A, B, C above) as

[ik"_l,iklt ] , and suppose if possib1e that f has

retrogression > 28 there: say, that f(ik"_l) > f(ik,,)

and that f(j") - f(j') > 28 for ° <Jo'<Jo"<i 1 k"_1 = = k"

Actua11y, the subfunction being critica1, f(ik"_l)­

f(j ') and f(j") - f(ik,,) are both ~ f(j") - f(j')

(which forces, incidenta11y, and Jo" ..L ° ) T 1 k " .

Let, as before, min If(ik ) - f(ik_1) I be attained for

k = k' • Of course f(ik"_l) - f(ik,,) > 28 , so k":; k'.

Th en {i 0 ' i 1 ' • . • , i k" -1 ' j , , j " , i k", i k" + 1 ' . . • , i n+ 1 } \ Ü k ' -1 '

i k ,} is a subfunction which, as compared to {io ,i1 , •.. ,

44 C. Davis

i n+l } , has lost at least one difference If(ik )­

f(ik_l ) I which is exactly 28 and has not gained any.

In view of the Convention of Section 2, this contradicts

the choice of {io,il, .•. ,in+l } as having maximal

discrepancy. (Remember to verify the assertions of

this paragraph even in the exceptional cases that

k' is 1 or n + 1 , and that ktf is k' - 1 or

k' + 1 .)

A very similar argument disposes of an interval of

monotonicity obtained by Case B or C above as

[ik '_2,ik, +l ]

Still to be excluded are the possibilities that

f has retrogression > 28 on IO,hl ] or Ihm_l,N]

Suppose if possible that f(j') - f(j") > 28 for

o < j' < j" ~ hl . The reasoning already given covers

the case 'I > h 'd J = 0' so cons~ er j 1 < h o

loss of generality assume that max {f(j)

attained at j = j' , and that min {f(j)

Without

, < '''} J = J is

is attained at j j". Similar conventions in treating

the alternative h <j'<j". m-l

We can now rule out the possibility that j" < h o

A Gombinatorial Problem in Best Approximation

(and its symmetrical variant

j" < h o

with f(j") > fCh ) o

j' > h ). m

Indeed, if

, then we are free to re-

45

define j" to be h o

(possibly redefining j' as well,

but so what?). If j" < h with f(j") < feh ) , then o 0

h can not be i , and neither can it be the j of 000

Gase B, else the criticality of the initial subfunction

would be violated; the remaining possibility is Gase A

with ho = i l ' and then {j',j",i2,i3, ... ,in+l } shows

that {io,il, ••• ,in+l } could not have had minimal

discrepancy.

Similarly, and more simply, we rule out the

possibility that ho < j" ~ hl (and its symmetrical

variant hm_l ~ j' < hm).

Gase G is now disposed of: if, say, f(j') - 28 >

f(io ) < feil) then the conditions for Gase B would pre-

vail, and hence Gase G would be excluded by definition.

All possibilities under Gase Aare obtainable by

symmetry from the following two:

j" = h = i • The manner of fixing j' and j" , o 1

together with the criticality of the given subfunction,

46 C. Davis

entail f(j') = f(i ). But h could not have been o 0

chosen as i l unless Ifeil) - feio) I = 20 , contradic-

ting f(j') - f(j") > 20

A2 •• h • o = 1.1 ,

is a subfunction which, as compared to {io,il, ••• ,in+l}'

has lost one difference If(ik ) - f(ik_l )I which is

exactly 20 and has not gained any - a contradiction.

In Case B, considering by symmetry only the

alternative h < h = i there are still two o 1 0'

possibili ties :

BI: j' < j" = h o

Then {j',h , •.• ,h} is a subfunc-o n

tion which is readily seen to violate the assumed

maximal discrepancy of {io, •.• ,in+l}.

j" > j' =

same role.

h n

Then {h , ••• , h ,j 11 } o n

will fill the

This completes the proof that IIf - gll ~ 0 •

Together with the Proposition, it supplies all that

was lacking to establish the first Theorem. Furthermore,

the function g constructed has all the special

A Gombinatorial Problem in Best Approxim&tion 47

properties prescribed in the second Theorem. (If, say,

i = 0 f k' - 1 , then h o 0

o , and gei ) o

does indeed

equal f (i ).) o

The structure of the argument may be brought out

by some examples.

It emerged in the proof that the oscillation order

of the best approximant g can always be taken to be

n - 1 or n. It can not always be taken to be n .

ExampZe 3: N = 3, n = 2

j o 1 2 3

f (j) o 2 1 3

Then every best approximant of order < 2 must have

order 1.

Thus Gase G is indispensable, and so, obviously,

is Gase A. As to Gase B, consider

Examp Ze 4: N = 5, n = 3

j o 1 2 3 4 5

f (j) 1 3 1 2 o 2

The only critical subfunctions of order 4 are of course

48 C. Davis

{O,1,2,3,4} and {1,2,3,4,S}. App1ying the ru1e of

Case C to either of these yie1ds an approximant at

distance 1 from f. We need the ru1e of Case B to

get {ho ,h1 ,h2,h3 } = {O,1,4,S} and IIf - gll = 1/2

Let me comment also on the Convention concerning

"maximal diserepancy". It recommends itse1f on general

grounds: it makes the choice of {io , ... ,in+1 } insensi-

tive to sma11 perturbations in the va1ues of f. But

it is also essential for the truth of the second Theorem!

ExampZe 5: N = 8, n = 4

j

f (j)

o

1

1

2

2

1

3

2

4

1

S

3

6

1

7

2

8

o

The subfunction {O,1,2,3,4} satisfies all requirements

of the Theorem except for the Convention, yet it gives no

e1ue to the best approximation. Less drastic is

ExampZe 6: N = 4, n = 2 .

j

f (j)

o

o

1

2

2

1

3

3

4

2

The subfunction {1,2,3,4} satisfies all requirements

of the Theorem except for the Convention, yet it leads to

A Combinatorial Problem in Best Approximation

a best approximant only if the choice of k' is for­

tunate.

4. THE SEARCH FOR THE SUBFUNCTION

Do the ideas introduced really help solve the

Problem? How would one actually set about finding 0

for a non-trivial instance with N around 100 and n

around 8?

49

One can write an algorithm which starts with an

arbitrary g of the required oscillation order and

progressively diminishes Ilf - gll , ending after a finite

string of changes with a best approximating g like that

constructed in the last Section. Inevitably, it uses the

same ideas as I have been using. There are no numerical

experiments of such a size as to be interesting; however,

I would guess it is ordinarily better to approach the

problem from the other side, as follows.

Setting o(n) for the 0 of Section 2, to make

explicit its dependence on n we note that 20(N-l) =

min. If(j) - f(j-l) land ask for a simple way of going J

from any o(n) to o(n-l) •

LEMMA: If {i , ... ,i } o m

has maximaZ disarepanay 20 "

50 C. Davis

then f has retrogression < 28 on eaah intervat ==

[ik_1,ikJ (k == 1, ••• ,m) •

Though we did not need this fact in Section 3, its

proof was essentia11y given there, so I will say no more

about it.

DEFINITION: If i o < i 1 < ••• < im 3 then to 'throwout'

a pair (ik_1 ,ik) will mean to replaae the set

{i , ... ,i} by o m

{i, ... ,i 1}3 if k=m; o m-

DEFINITION: Calt a subfunation {i , ... ,i } o m 'basia'

in aase it has maximal disarepanay 28 and3 beside

being aritiaal3 satisfies

1 f (j) - f (i )1 < 2n o =

(j < i ) o

(j > i ) m

('the endpoint aonditions').

A Combinatorial Problem in Best Approximation

THEOREM: From a basic subfunction {io '· •• ,in+l } with

discrepancy 28, throw out a pair (ik'_l,ik ,) such

that If(ik ,) - f(ik'_l)I = 28. If this is not enough

to determine k' , choose it so as to reduce by as much

as possible the number of steps If(ik ) - f(ik_l ) I

equal to 28. Then the resulting subfunction will

again be basic.

51

PROOF: It is evident that it is a critical subfunction.

The endpoint conditions are also easily checked. In

studying the alleged maximal discrepancy, we have to

compare with an arbitrary competing critical subfunction

{h ,hl, •.. ,h} of the same order m ; here m = n if o m

k' is 1 or n + 1 , and otherwise m = n - 1 •

When can there be an hk which is not among the

i l ? As a representative case, assume hk and i l

both maxima of the respective subfunctions, and hk_l <

Because both subfunctions are critical,

i l lies in an interval where f assumes its maximum

at ~ and also hk lies in an interval where f

assumes its maximum at i l . Therefore f(hk) = feil)

To assign to hk the new value i l would therefore not

52 C. Davis

change the discrepancy we are investigating. Nor wou1d

it spoi1 the relation

needed for critica1ity; because if any j( E ]il,~I

satisfied f(j') < f hk+1) then the retrogression of f

on Ii1 ,il +1] wou1d be greater than

which wou1d contradict the Lemma.

Another case, with hk and i 1 both maxima, is

i 1 ~ hk_2 < hk ~ i 1+1 • The Lemma shows that it can

occur on1y if

= 20

and f(hk) - f(hk_1) = 20. And the preceding paragraph

shows that if it does occur then we are free to suppose

that are among h , •.• ,h • o m

A Combinatoria1 Problem in Best Approximation 53

as

The above reasoning does not e1iminate such a case

h < h = i . but the endpoint conditions with the o 10'

Lemma show that it can occur if

discrepancy {h , •.• ,h } = 25 o m

Summing up, there is no 10ss of genera1ity in

confining attention to competing subfunctions

{ho,.·.,hm} which are obtained from {io , ••. ,in+1 }

by throwing out some pairs of its elements, and then

(in case the discrepancy can be kept at the va1ue 25)

inserting new elements into some of the remaining

interva1s, inc1uding rO,iol and ]in+1 ,N] •

This proves the Theorem in the special case that

min If(ik) - f(ik_1) I is attained for on1y one va1ue of

k. In the contrary case, one more observation remains

to be made: Suppose that, in the process described, the

throwing out of a pair (other than the pair (ik '_l,ik ,» reduces the number of steps ]f(ik ) - f(ik_1)] = 25 by

more than one. Then each of the corresponding insertions

of new hk in the process must have increased it by at

least the same number, otherwise the subfunction initia11y

54 C. Davis

given would not have had maximal discrepancy. I hope

tbe idea here is sufficiently clear that the reader can

check this observation. From it, we see that throwing

out (ik'_l,ik) gives a subfunction satisfying the

Convention.

The Theorem comes close to answering the call for

a simple way of passing from any o(n) to o(n-l) •

Wbat it provides directly is a simple way of passing

from o(n) to either o (n-l) or o (n-2) • The most

pleasing feature is that at each step one refers only

to the subfunction at hand, not to values of f at

any other points. One simply works down from basic

subfunction to basic subfunction, throwing out one

interval at each step, until n is small enough

or until o(n) is too large, if you prefer. At the

end, if it is desired to specify a best approximant,

there is a short computation on the whole domain

{O,I, ... ,N} .

A natural additional assumption (resembling the

hypo thesis of general position in geometry) is that a

non-zero value If(jl) - f(j2)! can equal !f(j3)-

f(j4)! only if the pairs {jl,j2} and {j3,j4} are

the same. Under this assumption, things are still

simpler, and a still more complete description can be

A Combinatorial Problem in Best Approximation

given:

(1) There is abasie subfunetion of a given order if

and only if the eritieal subfunetion of maximal dis­

erepaney is unique.

(2) Non-uniqueness arises only in the following way:

55

If {io, .•. ,in+l } is basic, and mink If(ik) - f(ik_l ) I is attained for k r l,n+l , then the eritieal subfunc­

tions of order n having maximal discrepancy are

{io,···,in } and {il, .•• ,in+l }.

Proofs are similar to the foregoing.

Given a function not satisfying the assumption,

one way to proceed would be to modify it slightly so

that it did, then deal with this simpler situation.

5. A PROPOSEn EXTENSION

The following generalization of the problem treated

here seems much harder.

PROBLEM: Given v functions fl, .•• ,fv on {O,l, ••• ,N},

to find v functions which wiZZ minimize

56 C. Davis

Hf - g//= max /f (j) - g (j)/ j K K

,K

subjeat to the aondition that no Zinear aombination of

the gK have osaiZZation opdep > n •

REFERENCES

1. Bartho1omew, D.J.: A test of homogeneity of means under restricted alternatives, J. Royal Statist. Soc. Sero B23(1961) , 239-273.

2. Davis, C.: Mapping properties of some Cebysev systems, Dok1. Akad. Nauk SSSR 175(1967), 280-283 = Soviet Math. Dok1. 7(1966), 1395-1398.

3. Kantorovic, L.V. and Rubinstein, G.S.: Aspace of comp1ete1y additive functions, Vestnik Leningrad. Gos. Univ. 13(1958), no. 7, 52-59.

4. Krein, M.G.: The ideas of P.L. Cebysevand A.A. Markov in the theory of 1imiting va1ues of integrals and their further deve1opment, Uspehi Mat. Nauk 6(1951), no. 4(44), 3-120 = Amer. Math. Soc. Trans1. (2) 12(1959), 1-121.

5. Videnskii, V.S.: An existence theorem for the po1ynomia1 with a given sequence of extrema, Dok1. Akad. Nauk SSSR 171(1966), 17-20 = Soviet Math. Dok1. 7(1966), 1395-1398.

GOOD APPROXIMATION BY SPLINES WITR VARIABLE KNOTS

* Carl de Boor

57

Consider approximation of a given funetion f, on

[0,1] say, by elements of Sk , i.e., by poZynomiaZ 1T

spZines 0/ order k (or, degree < k) on some partition

(t.)N+l of [0,1], ~ 0

Rere,

o = = 1 .

tl, ... ,tN are the knots or joints of k

SES , 1T

and their multiplieity, i.e., equality among two or

more of these, indieates redueed smoothness at that

knot loeation in the usual way.

Best approximation to f by elements of

* This work was supported in part by NSF grant GP-07l63

58 C. de Boor

quite we11 understood for a variety of norms since,

after all, Sk is a Zinear space. It seems practica11y 1T

more important and theoretica11y more interesting to

investigate the approximation to f by spZines of opdep

k with N knots, Le., by elements of

where the union is taken over all partitions 1T of

[0,1] with N interior points. For, the approximation

power of sp1ine functions seems to 1ie preciae1y in the

possibi1ity of p1acing the knots in a usua11y quite non­

uniform way to suit the pecu1iarities of the given f •

Yet the straightforward approach, vize the construction

k of a best approximation to f in SN' has turned out

to be beset with difficu1ties. It being a somewhat

nasty nonZinear minimization problem, no satisfactory

characterization of a best approximation can be found

in general, see e.g., [4] for the case of Chebyshev

approximation. Consequent1y, any computationa1 scheme

has to be content to find, by some descent method, a

ZoaaZZy best approximation, and even that seems to be

computationa11y quite expensive. Also, the function

f may be "given" in a way that makes the ca1cu1ation

of best knots impossible simp1y because 11 f - s 11

Splines with Variable Knots

cannot be calculated. E.g., f may be the unique

solution of some differential equation

m m-l D f(t) = F(t,f(t), •.• ,D f(t», for t E rO,lJ

with side conditions S.f 1

i=l, .•• ,m

where F, the linear functionals SI' ..• ,Sm and the

numbers cl, .•. ,cm are known, but the value of f at

t is not.

59

For these and other reasons, it becomes important

to search for methods which will produce relatively

cheaply good, if not best, knots for the approximation

of a given function from a variety of information about

this function. And the literature concerning bounds on

seems to be a good place to start such a search.

One approach, taken, e.g., by Freud and Popov [7],

[8], and by Sendov and Popov 116], has been to reduce

the problem of estimating dist(f,S~) to the simpler

problem of estimating dist(g,S~) for given g E eID,l]

60 e. de Boor

making use of a fact such as the fo110wing

LEMMA 1. Fop evepy f € e(k-2)[0,1] , and evepy papti­

tion ~ = (t.)N+1 fop [0,1], ~ 0

(1)

with max.ßt. ~ ~

A simple proof of this 1emma goes as fo110ws: On

e[O,l], define the linear map P by

P e[O,l] + Sk f + ~ f( )N l.i T. • k ~ ~ ~,

B-sp1ine basis for with (Ni,k) the norma1ized

(see, e.g., [3]). Since the Ni,k are nonnegative and

add up to 1 at any particu1ar point, it then fo110ws

that

I(f - Pf)(t)1 = ILi(f(t) - f(Ti»Ni,k(t) I

~ max {If(t) - f(Ti) I INi,k(t) # O} •

On the other hand, since Ni,k is nonzero on1y on

(ti,t i+k) , it is possib1e to choose T. ~

in [0,1] so

Sp1ines with Variable Knots

that

for all t E [0,1] , N. k(t) :f 0 1.,

With such a choice, one then obtains

Wf being the modu1us of continuity of f; hence, for

f E C(l)[O,l] , and arbitrary s E Sk ,

therefore

making use of the facts that

and repeated app1ication of this last inequa1ity gives

the estimate (1).

Choosing now, in particu1ar, TI so that

61

62 C. de Boor

and then augmenting TI by at most N - 1 points to

insure that

ITII ~ l/N ,

one obtains from (1) the estimate, valid for

f E c(k-2)rO,1]

The simpler problem of best approximation by

broken 1ines, i.e. in 2 SN ' is taken care of by a

resu1t such as the fo11owing

LEMMA 2. For every g E AC with g' E BV

This can be found, e.g., in [17] as a special case

of a much more general resu1t, but can also be proved

direct1y as fo11ows: If the straight 1ine s inter­

po1ates f at the points a < b , then

Splines with Variable Knots

(f - s)(t) = f[a,b,t](t-a) (t-b)

= flb,t]-f[a,t](t_a)(t_b) b - a

63

with f[ro, ••• ,rk] denoting the k-th divided difference

of f at It follows that

sup ](f-s)(t)] < (b-a)/4 Osc[a,b]f' a<t<b

b < (b-a)/4 f ]df']

a

if f E AC and f' E BV , where

Osc[a,b]g = ess.sup g - ess.inf g • [a,b] [a,b]

Hence, if such f is approximated by the broken

2 which interpolates f o = t s E SN_l at < t l 0

li ne

< ... < t 1 < N- t = 1 N , and the t i 's are chosen so that

for some a, then

64

while

(1 - 0) Var(f') N2

C. de Boor

by Jensen's Inequality, sinee I/x is eonvex for

x > o. This proves the lemma.

The two lemmata have the desired

COROLLARY. If f e C(k-2) [0,1] ~ with f(k-2) e AC

and f(k-l) e BV , then

(3)

This is to be eompared with the eustomary statement

that

(4)

in ease f e C(k-l) [0,1] with f(k-l) e AC and

Splines with Variable Knots 65

But, although this improvement of (3) over (4) was

achieved by a particular choice of knots, the argument

has to be suspect since it relies on choosing the knots

so as to produce a good approximation to f(k-2) rather

than to f. Even the more sophisticated argument of

Subbotinand Chernykh [17] (who obtain (3) by construc­

ting an approximation to f in the spirit of Birkhoff's

local spline approximation by moments [1], [2], followed

by an appropriate choice of knots so as to make the

error small) excludes consideration of such practically

interesting functions as

f(t) = t a , some a E (0,1)

and therefore does not give, e.g., Rice's startling

result [10] that

(5) for f(t) = t a with 0 < a ,

k -k dist (f,SN) < const kN

00 a,

Rice's argument is a direct verification that for

a certain set of knots selected according to a rule de­

pending on a everything works out. In an attempt to

66 c. de Boor

genera1ize Rice's resu1t, H. Burchard [5] proved the

fo11owing intriguing

THEOREM 1. Por f E c(k)[O,1J , and N ~ Nf ' and tor

where

er = er p,k 1/(k + 1/p) •

Simi1ar results have been obtained, for the special

case p = 2 , by Sacks and Ylvisaker [12-15], and more

or 1ess by McC1ure [9], again dealing only with

f E C(k) or even f E C(k+l) , and therefore not giving

Rice's resu1t (5). Nevertheless, these considerations

bring out the importance of the er-norm of f(k) for

a < 1 in the discussion of distp(f,S~) and suggest

that, e.g., (5) ho1ds because, for a f(t) = t

is finite. This is confirmed by the fo11owing

8p1ines with Variable Knots 67

THEOREM 2. Ir f E C[O,l] n c(k)(O,l] , and If(k)(t) I

is monotone deopeasing3 then

This can be proved as fo11ows: Consider approximation

to f k

in 8rr ' where rr has N - 1 distinot points

in (0,1)

say, but each repeated k times. Then

with Tf the piecewise po1ynomia1 function of order k

which, on (ti ,ti+1) , agrees with the Tay10r series for

f around up to terms of order < k , hence, for

1 1f t (k) k-1 I !f(t) - Tf(t)! = (k-1)! f (r)(t-r) dr t i +1

68

1 < -:( =-k -""';l'-:-)-:"'"!

1 .2k!

C. de Boor

The last inequality is easy to prove if, as we assume,

!f(k)! decreases monotonely (- consider both sides as

a function of t i+l and differentiate -) but impossible

to find in the literature. In any event, choose now

the t i 's so that the ßi 's defined above are all

constant,

hence

ß. = ß for all i. Then 1

1 1 1 -k 1 (k) 11 IIf - Tf L.2k! ß = k! N If l/k

which proves the theorem in view of the fact that

distoo(f,S~) decreases with increasing N

A similar result holds for distp(f,S~) with

p < 00 , as proved by D.S. Dodson in [6], where on can

also find the following

Splines with Variable Knots 69

THEOREM 3. If~ foT' some oonveT'ging net (7T) of par>ti­

tions of [0,1] ~ the oOT'T'esponding ZoweT' Riemann sums

foT'

1 J If(k)(r)10dr with ° l/(k+l/p)

o

oonveT'ge to A ~ then

foT' some positive oonstant constk independent of (7T)

and f •

These facts and arguments suggest that in approxi-

mating f by elements of k SN ' one should choose the

N knots tl, •.. ,tN so as to make

approximately constant as a function of i. This has

been tried by Dodson 16] in ascheme for the adaptive

solution of an ordinary differential equation. From

a current piecewise polynomial approximation of order

< k to the solution f , he guesses a piecewise

70 C. de Boor

constant approximation g to f(k) , and then se1ects

a new knot set so as to equa1ize over

subinterva1s. To give an examp1e, Russe11 and Shampine

[11] solve the problem

ei"(t) - (2-t2)f(t) = -1 on [-1,1]

with f(-l) = f(l) = 0

for E = 10-8 by co110cation, using sp1ines of order

6 with 47 distinct knots, each of multiplicity 3. The

knots are p1aced on an ad hoc basis so as to pile up

near + 1. They obtain an approximation with error

-6 -4 of 10 near zero, deteriorating to an error of 5.10

near the boundary. Dodson obtains the same accuracy

with 19 distinct knots, and obtains, with 47 knots,

-6 an accuracy of 2.10 even near the boundary (and an

10-8 error in the midd1e of the interval).

REFERENCES

1. Birkhoff, G.: J. Math. Mech. 16(1967), 987-990.

Sp1ines with Variable Knots 71

2. de Boor, C.: J. Math. Mech. 17(1968), 729-736.

3. de Boor, C.: J. Approx. Thy 6(1972), 50-62.

4. Braess, D.: Numer. Math. 17(1971), 357-366.

5. Burchard, H.: "Sp1ines (with optimal knots) are better" , to appear in J. App1icab1e Math. 1 (1972).

6. Dodson, D.S.: Ph.D. Thesis, Comp. Sei. Dpt., Purdue University, Lafayette, Ind. (1972).

7. Freud, G. and Popov, V.A.: Studia Scient. Math. Hungar. 5(1970), 161-171

8. Freud, G. and Popov, V.A. : Proc. Conf. Constr. Thy. Fctns, Hungar. Acad. Sei. (1970) , 163-172.

9. McC1ure, D.E.: Ph.D. Thesis, Div. Appl. Math., Brown University, Providence, R.!. (1970).

10. Rice, J .R.: in "Approximations with special emphasis on sp1ine functions", !.J. Schoenberg ed., Acad. Press, New York (1969), 349-365.

11. Russe11 , R.D. and Shampine, L.F.: "A co11ocation method for boundary va1ue problems", to appear in Numer. Math. (1972/73).

12. Sacks, J. and Y1visaker, D. : Ann. Math. Stat. 37 (1966) , 66-89.

13. Sacks, J. and Y1visaker, D. : Ann. Math. Stat. 39 (1968), 49-69.

14. Sacks, J. and Y1visaker, D. : Ann. Math. Stat. 41 (1970) , 2057-2074.

15. Sacks, J. and Y1visaker, D. : Proc. 12th Bienn. Sem. Canad. Math. Conga 115-136.

72 C. de Boor

16. Sendov, B1. and Popov, V.A.: C.R. Acad. Bulgare Sei. 23 (1970), 755-758.

17. Subbotin, Yu.N. and Chernykh, N.I.: Matern. Zametki 7 (1970), 31-42.

73

SATURATION AND INVERSE THEOREMS FOR SPLINE APPROXIMATION

R. DeVore & F. Richards

The purpose of this note is to examine the con­

nections between the smoothness of a function and its

degree of approximation by algebraic polynomial splines

of a fixed degree. Results of this type are known,

usually in the form of an estimate for the degree of

approximation for a certain method of spline approxi­

mation in terms of the smoothness of the function.

Estimates like this are customarily called direct

theorems of approximation. Our main interest lies in

the opposite direction, i.e., what inferences can be

made about the smoothness of a function when its degree

of approximation is known.

We say S is a spline of degree k - I if there

are points 0 = Xo < Xl < ••• < xm = I such that on

each interval [xi_l,xi ), i = 1,2, ••• ,m, S is an

74 R. DeVore & F. Richards

algebraic polynomial of degree at most k - 1. The

points X. 1.

are called the knots of the spline. For

generality, we make no restrietion on the continuity of

S at the knots.

If 0 = {O = Xo < xl < ••• < xm = I} , let S(o)

denote the collection of all splines of degree k - 1

with knots contained in 0 Define the error in

approximating f by S(o) as

E 0 (f) = inf" f - s 11 , SES (0)

where 11.11 denotes the supremum norm on [0,1].

o n

Now suppose (0) n

= {O = x(n) < (n) o xl

is a sequence of sets of knots,

< ••• < x(n) = 1} • m

n We let

max l<i<m --n

I (n) (n) I xi - xi _l

and assume that 110 11 -+ O. This guarantees that n

Eo (f) -+ 0 for each f € C[O,l] • n

By ßk we denote the t

th k-- difference operator

Saturation and Inverse Theorems 75

th so that the corresponding ~ order modu1us of continuity

of f is given by

sup Illlk(f,x)II [0,1-kt] o<t<h t

The notation 11.II[a,b] is used to indicate that the

norm is taken over [a,b] When

the norm is understood to be over

[a,b] is omitted,

[0,1] •

A proof of the f0110wing direct estimate for

Eo (f) in terms of wk(f,h) can be found in [3] • n

THEOREM 1. Suppose 0 < a ~ k, f E C[0,1] , and

wk(f,h) = O(ha ) as h + O. Then

(1) Eo (f) = 0(11 on lIa) (n + (0) • n

We shou1d note that the estimate (1) can actua11y be

obtained by using sp1ines

k-2 SEC [0,1]. n

S E S(o ) , with n n

76 R. DeVore & F. Richards

Our main concern is in what sense are the estimates

of (1) the best possib1e? We ask the fo11owing two

questions: When does Eo (f) = O(lIon"a), (n -+00) imp1y n

that a

wk(f,h) = O(h ), (h -+ 0) , Le., does the inverse

theorem to (1) hold? Second1y, is it possib1e to

improve (1) if we assurne higher smoothness for f?

It is not possib1e to answer these questions with­

out some additional restrictions on the sets of knots.

The easiest way to see this is when a fixed point, say

1 2 ' appears in each set o n

Then any sp1ine S

which has a single knot at 1 2

will sa tis fy Eo (S) = 0, n

n = 1,2, ••• , but S need not even be continuous. More

generally, the same phenomenon manifests itse1f when a

fixed point on1y falls in sma11 interva1s, in comparison

to 110 11. In order to avoid this, we will require that n

(0) satisfies the fo11owing mixing condition: n

(2) There is a constant p > 0 with the property that

whenever

such that

n > 0 and 1 < i < m -1 ~ there - n (n') (n)

x. < x. (n' )

< x j +1 with J ~

is an n' > n

Saturation and Inverse Theorems 77

min (I x (n) _ x (n ') I I x (n) _ x (n ') I) > pli 0 11 i j , i j+1 n •

It is easy to see that (2) guarantees that the fo11owing

must hold:

(3) Thero is a constant p > 0 with the property that

whenever n > 0 and x E: [O,l-pllö 11] .J then thero is n

(n') (n') an n' > n such that x j < x < x j +1 with

Note that equa11y spaced knots (i.e., ö = (i)n) n n 0

satisfy the mixing condition.

If the mixing condition ho1ds then we can show

that the estimate (1) is the best possib1e in the sense

we have asked.

THEOREM 2. Let (0) be a sequenae of sets of knots n

whiah satisfy the mixing aondition (2) and 11 0 11 -I- 0 • n

If 0 < a ~ k and f E: C[O,l] .J then

(4) E 0 (f) = ° (11 on W), (n + 00) n

if and onZy if

78 R. DeVore & F. Richards

and

(5) Eo (f) = O(llonllk), (n -+ (0) , n

if and onZy if f is a poZynomiaZ of degree < k - 1 •

Remark: The theorem ho1ds without the restriction that

lIonll tends to 0 monotonica11y but the proof becomes

somewhat more cumbersome and hides the essential ideas

invo1ved.

The equiva1ence in (4) is the inverse theorem to

(1). The equiva1ence in (5) estab1ishes the saturation

phenomenon for sp1ines and shows that the estimates (1)

can not be improved by assuming higher smoothness for

the function. Since Theorem 2 is proved with no con­

tinuity requirement at the knots, it app1ies to any

sp1ine approximation method provided the mixing con­

dition on the knots ho1ds.

Theorem 2 is a1ready known for approximation by

sp1ines with equa11y spaced knots. K. Scherer [6] has

given a proof of this under the additional assumption

that the sp1ines are smooth (i.e., in Ck- 2 [O,1]).

Scherer's proof is based on the general method for

obtaining inverse theorems deve10ped by P.L. Butzer

and K. Scherer [1]. For the saturation parts of

Saturation and Inverse Theorems 79

Theorem 2 with equally spaced knots, independent proofs

have been given by D. Gaier [4] (a "0" theorem) and

F. Richards [5] ("0" and "0" theorems). Our proof of

Theorem 2 is new and quite simple and of course has the

additional advantage of handling non-equa11y spaced

knots. Also, our technique can be genera1ized to give

inverse and saturation theorems for Chebyshevian sp1ines

(see [2]).

PROOF OF THEOREM 2. Because of Theorem 1, we need only

estab1ish the necessity in (4) and (5). We will on1y

consider (4) since the proof for (5) is almost identica1.

First observe that because of the mixing condition (2)

and the assumption that (11 on 11) is monotone, we must

have

(6) 1,2, ...

Now, suppose that S E::S(o), n n

o < CI. < k , and

(7)

with K a constant. We want to show that wk(f,h) =

o (hCl.) , (h + 0) . Choose n so that

80 R. DeVore & F. Richards

(8)

We will on1y consider those h 's for which the index n

1 in (8) satisfies 11 on 11 < 4. This covers all

sufficient1y sma11 h •

Let 0 < t < h • 3 Then for any x E [0'4] , (8)

gives that

(9) [x,X+kt] =- [x,X+kh] =- [x,x+pllon ll] =- [0,1] •

Since (3) ho1ds, there is an n' > n satisfying

(10)

Now, S, n

is a po1ynomia1 of degree at most k - 1 on

(n') (n') [xi ,xi +1 ) Therefore,

Using (7) in the last expression, we find that there is

a constant K1 such that

Saturation and Inverse Theorems

3 x E: [0 '4] , 0 < t < h •

where in the second to last inequa1ity we used (6) and

in the last inequa1ity we used (8).

3 To get the inequa1ity (11) for x E: [4,1-kt] , we

consider the function g(x) = f(l - x). The sp1ines

T (x) = S (1 - x) have their knots contained in n n

6' = {l_x~n) x~n) E: 6} and satisfy n 1 1 n

81

The sequence of sets of knots (6') also satisfies the n

mixing condition (2). Hence, arguing as we have in

obtaining (11), we find that

This shows that (11) is valid for

o < t < h. Therefore, wk(f,h)

necessity in (4) is estab1ished.

o < t < h •

3 x E: [4,1-kt],

< K2h<X and the

82 R. DeVore & F. Richards

REFERENCES

1. Butzer, P.L. and Scherer, K.: Approximation­sprozesse und Interpolations-methoden. (Hochschulskripten 826/826a) Bibliograph. Inst. 1968, 172 pp.

2. DeVore, R. and Richards, F.: The degree of approximation by Chebyshevian sp1ines. (to appear).

3. Freud, G. and Popov, v.: On approximation by sp1ine functions, Proc. of the Conference on Constructive Theory of Functions, held in Budapest, 1969, 163-172.

4. Gaier, D.: Saturation bei Sp1ine-Approximation und Quadratur, Numer. Math. 16 (1970), 129-140.

5. Richards, F.: On the Saturation c1ass for sp1ine functions, Proc. Amer. Math. Soc., 33 (1972), 471-476.

6. Scherer, K.: On the best approximation of continuous functions by sp1ines, SIAM J. of Num. Analysis, 7 (1970), 418-423.

SATURATION CLASSES FOR EXPONENTIAL FORMULAE OF SEMI­

GROUPS OF OPERATORS

z. Ditzian & C.P. May

83

A C semi-group of operators T(t) on a Banach o

space B into itself can be approximated by formulae

known as exponential formulae [5, p. 359]. The rate

of convergence of some of the exponential formulae in

terms of the moduli of continuity of T(t)f and

AT(t)f ,where A is the infinitesimal generator, was

investigated in [1], [3], and [4]. In this paper we

shall find the optimal rate of convergence, that is,

the saturation sequence, and the class of functions

on which it is achieved, that is the saturation class

for exponential formulae satisfying certain conditions.

These conditions are satisfied by the exponential

This research was supported partly by NRC grant A48l6.

84 z. Ditzian & C.P. May

formu1ae of Rille, Kenda11, Post-Widder and Phi11ips.

Also, it is important to note that the resu1ts depend

on1y on points in (t,t+O) for some 0 (no matter how

small) •

2. PRELINIMARIES AND DEFINITIONS.

Many exponentia1 formu1ae, and actua11y all those

treated in this paper, can be written as:

(2.1) 00

S(T,t)f(·) = f W(T,t,u)T(u)f(·)du o

where T has either the va1ues in {Ti} (Ti + 0+) or

all va1ues in (O,n) , and W(T,t,U) is a positive

kerne1 satisfying the fo11owing:

00

(2.2) Um T-1{! W(T,t,u)du-1} = o , T+O+ 0

00

(2.3) 1im -1 f T W(T,t,U) (u-t)du 0 , T+O+ 0

Saturation C1asses 85

(2.4) 1im .-1 !W(.,t,u) (u-t)2du = pet) I 0, t > 0 , .+0+

and for any positive ö and real a

(2.5) 1im .-1 ! W(.,t,u)eaudu = 0 . • +0+ I u-t I~o

Remark 2.1. It is easy to see that the assumption

CX>

(2.6) ! W(.,t,u) (t_u)4eaudu = 0(.) o

• + 0+ for any real a

imp1ies (2.5). For our particu1ar purposes (2.6) is

eas1iear to verify.

We define the c1ass of functions B2 (t) as

(2.7) B2 (t) :: {f E B, f E V(A2T(t+c:) for all E > O} •

Obvious1y we have for n > 0

(2.8)

86 z. Ditzian & C.P. May

~

n V(A2T(t+11» where 11>0

We define ~

V(A2T(t+11» is defined as the c10sure with respect to

B of V(A2T(t+11» , that is f E B such that for

every 11 > 0 a sequence {f }, n

exists satisfying 1im f = f n

in B and

~

Whenever B is reflexive, B2(t) = B2(t) (as

can be seen fo110wing [2, p. 373]).

3. THE SATURATION RESULT.

We sha11 state and prove the f0110wing saturation

resu1t which we will app1y in section 4 to various

exponentia1 formu1ae.

THEOREM 3.1. Let S«,t)f be defined by C2.1) and

satisfy (2.2) - (2.5), then for f E B

group T(t) of operators, we have:

and a C semi­o

(A) IIS«,t)f - T(t)fll = 0«)< -+- 0+ for aZZ t > t o

if and onZy if f E V(A2T(to» and A2T(to)f = 0

Saturation Classes

(B) IIS(T,t)f - T(t)fll = O(T)T + 0+ tor alZ t > t

PROOF. To prove sufficiency we recall that if

f E V(A2T(t1» , then for t 2 > t 1 T"(t )f 2 exists

and is equal to A2T(t2) . Therefore for any

t = t 2 > t 1 > t , and sorne o = o(t) > 0 , we have

for It-ul < 0

1 2 2 T(u)f - T(t)f = (u-t)AT(t)f + 2(u-t) A T(t)f

2 + E1(u-t) ,

where IIElll2.E. This irnplies for fE V(A2T(t1»

using (2.2),

2.IIT-1 f W(T,t,u){(u-t)AT(t)f lu-tl<o

122 2 + 2 (u-t) A T(t)f+E(U-t) }du-

- } P(t)A2T(t)fll

87

o

88 z. Ditzian & C.P. May

+ 11 T -1 f W(T, t,u){T(u)f-T(t)f}dull lu-tl >0

Recalling (2.5) and IIT(u) 11 ~ MeClu , 12 = 0(1) •

Combining (~.3) with (2.4) and the above, we get

11 = 0(1) •

Therefore one can easi1y see that f E B2(to)

imp1ies IIS(T,t)f - T(t)fll = O(T) for t > t ~ 0

1f f E V(A T(t1» , then a sequence {fn} exists

fow which fn E V(A2T(t» and 11 A2T (t1)fn 11 ~ M •

Therefore, for t > t 1 T"(t)fn exists for all n

and 11 T" (t) fn 11 ~ M1 ' which imp1ies

Since S(T,t)f (for a fixed T) and T(t)f are

continuous as operators on B, the sufficiency of

the condition in (B) is proved whi1e that of the

condition in (A) is simpler.

Saturation C1asses

To prove that eonditions in (A) and (B) are

neeessary too, we introduee first the operator

2 2 2 l/n l/n fami1y J • J = n f f T(u+V)dudv (see also

n n o 0

[2, p. 502-506]). J2 eommutes with T(t) and n

therefore with S(T,t) •

Fo110wing known eonsiderations [2, p. 505],

J2 f E V(A2), J2 f E V(A2T(t», n n

A2J 2f = n2(T(1)_I)2f n 2

89

and (JL)2T(t)J2f = A2T(t)J2f . dt n n Therefore using (3.1)

on J2f (instead of f) for t > t n 0

1 2 2 -1 2 2 -2 p(t)A T(t)Jnf = s - 1im T [S(T,t)J f - T(t)J f] T~ n n

= s - 1im J 2{T-1 (S(T,t)f - T(t)f} • T~+ n

2 Sinee A is a e10sed operator (see [1, p. 11] for

examp1e) and s - 1im J2f = f , we obtain (in ease n

(A», fo110wing [2, p. 505], f E V(A2T(t» and

A2T(t)f = 0 for all t > t However, this implies o

90 z. Ditzian & C.P. May

T(t )f + T(t +2h)f - 2T(t +h)f for all h ,0 0 2 0 = A2T(~)f

h

= 0 or A2T(t)f = 0 . o

For case (B) we obtain, fo11owing 12] again,

f E A2T(t) , but since this is shown for all

the proof is comp1eted.

t > t o

4. APPLICATIONS TO VARIOUS EXPONENTIAL FORMULAE.

In this section we sha11 state the resu1ts that

fo11ow Theorem 3.1 for various exponentia1 formu1ae.

First we denote the exponentia1 formu1ae as fo11ows:

(4.1) Sl(T,t)f 00 k

= e-t / T L ~, T(kT)f k=o .

for t > 0;

Saturation Classes 91

for t > 0

00 00 2 n n-l = e-At{f e-Au l (A t) U T(u)fdu+f}

o n=1 nl (n-l) I

for t > 0 ;

for t > 0 ;

00 n (4.6) S6(L,t)f = r (t-~) ~nT(a)f

n. L n=o

00 n n = l (t-a) l (_I)n-k(n)T(a+kL)f

n k n=o nIL k=o

for t > a > 0 •

92 Z. Ditzian & C.P. May

We sha11 denote our exponentia1 forrnu1ae Si(T,t)

i = 1,2, ••. ,6 understanding that T for i = 2 or

3 attains on1y va1ues 1 n

(or a subsequence of l) n

THEOREM 4.1. Let f E Band T(t) be a C semi o

gpoup on a Banaoh spaoe B, then fop t > 0 , 0-

A: 11 Sk (T , t) f - T (t)f 11 = 0 (T) fop t < t < t+o 0 0

fop some k if and onZy if f E V (A2T (t )) 0

A2T(t )f = 0 • 0

B: 11 Sk (T , t) f - T (t)f 11 = 0 (T) fop t < t < t +cS 0 0 ~

fop some k if and onZy if f E B2 (to) •

PROOF. The proof is actua11y simp1y verifying (2.2) ,

(2.3) , (2.4) and (2.6). For 1 S3 (u' t) and 1

S4 (-, t)

the estimates are to be found in [4]; other estimates

are also readi1y avai1ab1e or can be ca1cu1ated in a

re1ative1y simple, but somewhat tedious, way. The

function pet) in (2.4) which we will denote as

Pi(t) for Si(T,t) is given by:

Saturation C1asses

(In fact, after some ca1cu1ations, one obtains

S6(T,t) = exp «t-a)AT)T(a) which reduces S6(T,t)

to S1(T,t) operated on g = T(a)f .)

Remark 4.2. In our theorem we actua11y proved that

IISk(T,t)f - T(t)fll = O(T) (or O(T»

for t < t < t +0 o 0

are equiva1ent for various k for which Sk(T,t)

is defined for t E (t ,t +0) • o 0

Remark 4.3. If {T.} is a sequence of rea1s ~

is enough to obtain the necessary and sufficient

condition in (A) (or (B» since we did not use in

Theorem 3.1 more than the va1ues of S(T,t) on a

sequence T. + 0+. ~

93

94 Z. Ditzian & C.P. May

5. COROLLARIES AND REMARKS.

(I) It is tempting to try and follow the proof in

[2, p. 50] and prove Theorem 3.1 or 4.1 using the

estimate of "Sk(T, t ) - T(t )f 11 on1y, that is, at o 0

one point t o

The proof, however, does not fo1low

since neither

T"(t ) o

nor S(t) = s - 1im JL [T(t -h)-2T(t )+T(t +h)] h2 0 0 0

which are 1ike1y to rep1ace A in [5, p. 505], are

c10sed operators. This can be seen using the

examp1e: T(t)f(x) = f(x+t),

f(x) = {~ - x o < x < 1 x > 1

and

{ 1-~-X f (x) = n 0

o < x < 1

x > 1 1-­n

f E C , o

1 n , for which

S(1)f (x) = T"(1)f (x) = 0 while S(l)f and n n

T" (1) f do not exis t.

(11) In fact, the difficu1ty expressed in (I) is

not on1y in the proof since an example for which

Saturation C1asses

but 1im" 12 {T(t-h)f-2T(t)f+T(t+h)f}II = co h

can be given as fo11ows:

Let S(. ,t) r

B = C [O,co), o

T(t)f(x) = f(x+t) and f(x) be given by

o ,

00

and f(x) = l f (x) m=3 2m

otherwise

2r 1 Obvious1y, when h = 2- and t = 2'

95

T(t)f = T(i + h)f = 0 and ,,~ (T(t-h)f-2T(t)frT(f+h)fll h

2r+1 1 2r > 2 -- = 2 which is not bounded. We have

2 _2r

96 z. Ditzian & C.P. May

where 11 , 1 2 and 13 are the sums on 3 ~ m < r,

m = rand m > r respective1y.

k1 2r+1 r+1

11 1 I \ (2 k ) 1 22 (k '\ r~l I 11 1 ~suxp k~o (2)·T r+1J l.. f m(x)

22 m=3 2

-1 _2 r - 1 -3 2r - 1 2r+1 where k1 = max {klk ~ (2 -2 +2· )2 }

or, since IIrI1

f m(x) 11 ~ 1 , we have, fo11owing [6; m=32

p.1S (8)] for 1

IX = 4'

11 1 3 11 ~ SUpIS2(S-2r+1,~) I f m/ ~ M sup I If ml x m=r+1 2 x m=r+1 2

2r+1 ~2M2- ,(M= sup IIT(t)lO •

t<l

Saturation C1asses

-1 _2r -3 2r 2r+1 where k2 = max {klk ~ (2 -2 +2 ·)2 } and

supp f ( ~1 + x) are disjoint, and therefore 2r 2

2

2r+1 2r+1 r r 2 "I~I~ e k ) (t)2 ·2-2 2. C(2-2) •

2

(111) For ho10morphic semi-groups we a1ways have

"Sk(T ,t)f - T(t)fll = O(T). In this case

IIS(T,t )f - T(t )fll = O(T) imp1ies, using (3.1), o 0

that A2T(t)f = 0 . o

(IV) If T(t) is a group of operators such that

s - 1im T(t+h) = T(t) , then A = s - 1im T(h)-I h+O h+O h

is a c10sed operator and so is A2 • In this case

11 Sk (T ,t )f - T (t )f" = 0 (T ) if and on1 y if o 0

2 = 0 or A f = 0 ,

97

98 z. Ditzian & C.P. May

~d

Ilsk("to)f - T(to)fll = 0(,) if and on1y if

f E V(A2T(t » = V(A2) • o

Such a situation occurs when B = Co(_~,m) ~d

T(t)f(x) = f(t+x) •

REFERENCES

1. Butzer, P.L. and Berens, H.: Semi-groups of operators ~d approximation, Srpinger-Ver1ag, 1967.

2. Butzer, P.L. ~d Nessel, R.J.: Fourier analysis and approximation, Vo1. I, Birkhauser-Ver1ag, 1971.

3. Ditzi~, z.: On Hi11e's first exponentia1 formu1a, Proc. Amer. Math. Soc., 22 (1969), pp. 351-355.

4. Ditzian, Z.: Exponentia1 formu1ae for semi-group of operators in terms of the reso1vent, Israel Jour. of Math., Vo1. 9, #4, 1971, pp. 541-553.

Saturation C1asses 99

5. Hi11e, E. and Phi11ips, R.S.: Functiona1 analysis and semi-groups, Americam Mathematica1 Society Co11oquium Pub1ications, Vo1. 31, 1957.

6. Lorentz, G.G.: Bernstein Polynomials, University of Toronto press, 1953.

101

ON SOME CONJECTURES OF ASKEY CONCERNING COMPLETELY

MONOTONIC FUNCTIONS

J.L. Fields & M.E. Ismail

INTRODUCTION.

In [1], Richard Askey analysed the LP conver­

gence of the Lagrange interpolation polynomials when

the zeros of the classical Jacobi polynomials,

p(a,ß)(z) , are used as the points of interpolation. n

His analysis was complete, except for some results

concerning the positivity of the Cesaro means of some

order y, (C,y) , for the Poisson KerneI,

1 K = f

n -1

1 a,ß ~ - I' 0 < r < 1 •

102 J.L. Fields & M.E. Ismail

Taking into account iso la ted results of this nature,

(e.g. Fejer [4] showed that for

(C,2) means of P (x,y) r

are positive, while

the

Kogbetliantz [10] showed that for 0. = ß , the (C,2a.+2)

means of P (x,y) are positive) Askey [2] made the r

following conjecture.

CONJECTURE 1.

positive for

The (C,a. +ß+2)

-1 a.,ß ~ '2 .

means of P (x,y) r are

See [2] for a more complete list of when this and the

following conjectures are true.

Using some results of Gasper [8, 9] on convolution

structures, Askey pointed out that it is actually

sufficient to prove Conjecture 1 with y = 1. Thus,

Conjecture 1 is equivalent to the following.

CONJECTURE 1'. If A = a.+ß+l , and

G(t, z) -().+1) ( ),+1, ),+2

= (l_t2) F 2 2 2 1

ß + 1

()C)

= l: g (z)tn n n=o

4zt ) (l+t) 2

Completely Monotonie Funetions

then

g (z) > 0 n

o < z < 1

103

Assuming the validity of Conjeeture 1 (or 1'),

Askey [2] dedueed several other related results.

Stated as eonjeetures, they are as foliows:

CONJECTURE 2. If

H(t,z) = (1_t)-2Ä[1+2t(2z_l)+t2]-Ä ,

then

co

= L h (z)tn n n=o

h (z) > 0 n

o < z < 1 Ä > 0 •

Remark: This eonjeeture is known to be true for

Ä = t and follows from Kogbetliantz's for Ä > 1 •

CONJECTURE 3. If 4y = 2a+3 , then fop x > 0

104 J.L. Fie1ds & M.E. Ismai1

1 --4y () 22 r (2y+1)x6y- 1 2y I 2

r(n+1)r(6y) 1F2 1 -~ 3y,3Y+2

~ 0, 2n > -1 or 2y ~ 1 •

Remark: This eonjeeture is known to be valid for

2y = 1,2, ••••

CONJECTURE 4. If 2y = 2n+1 , then fop x > 0

o

( y I 2) -x 1'4 '

2y, 2y~

Remark: Quite reeent1y, Askey and Po11ard [3] have

estab1ished this eonjeeture for y > 0 •

By Bernstein's Theorem on eomp1ete1y monotonie

funetions, Conjeetures 3 and 4 are equiva1ent,

respeetive1y, to the fo11owing:

CONJECTURE 3'. x-2Y (x2+1)-2y is eompZeteZy monotonie

Comp1ete1y Monotonie Funetions

fop 2y ~ 1 and x > 0 •

CONJECTURE 4'. x- 2Y (x2+1)-Y is aompZeteZy monotonia

fop y > 0 and x > 0 •

In Theorem 1, we prove a resu1t whieh ine1udes

Conjeetures 2 and 4. Conjeeture 4 is further genera-

1ized in Theorem 2. Conjeetures 3 (and 4) ean be

proved using asymptotie methods deve10ped in [5], but

the proofs are 10ng, and will appear e1sewhere.

THEOREM 1. If

then fop aZZ n,

p (z) n

r(n+2pA) (-n,n+2PA,AI ) = -r"""'(n-+;:""l-:-)-r'-:-( 2""'::P-A"'-) 3 F 2 1 I z ,

PA ,PA+z

> 0 P > 2, A > 0, 0 < z < 1 .

and

105

106 J.L. Fields & M.E. Ismail

lim nl-2PApn(~21 = ----,-1_.,- F r(2pA) 1 2 n-+o:l n

( A ) 1 -z , PA 'PHzl

> 0 p > 2 A > 0 z reaZ.

PROOF. The identification of p (z) follows from the n

general formula ([7])

A A+l

(2'-2-' a l , ... ,ap )

(l-t)-A F ! -4tw p+2 q 2

bl, ••• ,bq (l-t)

CX>

r(n+A)tn F (-n,n+A,al, ••• ,ap ! ) r(A)n! P+2 q b b w

1'· .. , q = L

n=o

q ~ p+l, Itl < 1 ,

and the identification

F(t,z) = (1_t)-2PA F (AI -4tz ) 1 0 (1_t)2

A trivial modification of the Askey and Pollard

proof [3] yields the non-negativeness of the p (z) • n

Completely Monotonie Funetions 107

Set

q(t) = -2(p-l) log (l-t) .- log [1-2t eos 8+t2] ,

l-2z = eos 8 •

Then

q' (t) 2(p-l) + 2 eos 8-2t = , l-t i8 -i8 (t-e )(t-e )

2(e-l ) -i8 i8

+ e + e = l-t -i8 i8 I-te I-te

00

= L {2(p-l) + 2 eos [ (n+I)8]}tn > 0 p > 2 • n=o

Thus q(t) is absolutely monotonie (q(O) = 0) ,

so is F(t,z) = eAq(t) , whieh implies p (z) > 0 n -

all n. The last statement of the theorem then

follows from the non-negativeness of the p (z) n

a result in [6] eoneerning eonfluent limits.

Remark 1: One might be tempted to try to prove

Theorem I for

from [7], that

3 p = - and 2

A > I , but it follows

and

for

and

108 J.L. Fie1ds & M.E. Ismai1

p (1) = 2-2AN2PA-2A-1{ 1 + O(N-2)} n r(2pA-2A)

+ (_1)n22A-2PAN2A-1 { 1 + O(N-2)} , r(2A)

which can be negative for p < 2 , imp1ying that p ~ 2

is a necessary condition for

on 0 < z < 1 .

p (z) n

to be non-negative

Remark 2: In [3], Askey and Po11ard show that the

p = 2 resu1t leads to a simple proof of the

Kobet1iantz resu1ts concerning the positivity of the

(C,2a+2) means of the Poisson kerneI.

A final genera1ization of Conjectures 2 and 4 is the

following.

THEOREM 2.

( -n,n+2pA ,p-a I )

3F2 z pA+b,PA+C

> 0 o < z < 1 .

F (A-a \_112) > 0 1 2 pA+b,PA+C 4

11 reaZ"

o < a < A, 0 < b 1 "2 2. c, p > 2 A > 0 •

Comp1ete1y MOnotonie Funetions

PROOF. Consider the Beta transform

g(t) (-n.n+2PA ;AI 2) = 3F2 zt

C,D o.+1,ß+1 > 0

With

g (t) rn .... 20A .A I 2) = 3F2 zt 0 1 p). ,p).~

o 2 t,z ~ 1

0.+1 = 2p)., ß+1 - b, b > 0 ,

one obtains

> 0

App1ying the above transform to

0. = 2p). ß + 0.+3 = p).+e , 2

o ~ t,z ~ 1 .

with

1 e >-2

109

110 J.L. Fields & M.E. lsmail

one obtains,

_ ( -n, n+ 2p ;\. ,;\. I 2) - 3F2 zt

p;\.+b,p;\.+c :> 0 , o ~ t,z ~ 1 •

Finally, if the Beta transform is applied to g2(t)

with

a + 1 = 2;\' - 2ß - 2 = 2;\' - 2a, 0 < a < ;\. ,

one obtains the first statement of the theorem with

strict inequalities. The general theorem is then

obtained by taking limits. The same sequence of

transformations yields the statement concerning the

lF2 'so This result can also be obtained by taking

the confluent limit of the 3F2 polynomials, see [6].

REFERENCES

1. Askey, R.: Mean convergence of orthogonal series and Lagrange interpolation, Acta Math. (Budapest), to appear.

2. Askey, R.: Summability of Jacobi series, to appear.

Comp1ete1y Monotonie Funetions

3. Askey R. and Po11ard, H.: Some abso1ute1y monotonie and eomp1ete1y monotonie funetions, SIAM Journal Math. Anal. to appear.

4. Fej~r, L.: Neue Eigenschaften der Mittelwerte bei den Fourierreihen, J. London Math. Soe. 8 (1933), 53-62, Gesammelte Arbeiten 11, 493-501.

5. Fie1ds, J.: A uniform treatment of Darboux's method, Areh. Rat. Meeh. and Anal., 27 (1968), 289-305.

6. Fie1ds, J.: Conf1uent expansions, Math. Comp. 21 (1967), 189-197.

7. Fie1ds, J.: Asymptotie Expansions of a C1ass of Hypergeometrie Polynomials with Respeet to the Order 111, J. Math. Anal. App1., 12 (1965), 593-601.

8. Gasper, G.: Positivity and the eonvo1ution strueture for Jaeobi series, Anna1s of Math. 93 (1971), 112-118.

111

9. Gasper, G.: Banach a1gebras for Jaeobi series and positivity of a kerne1, Anna1s of Mathematies 95 (1972), 261-280.

ID. Kogbet1iantz, E. Recherehes sur 1a sommabi1it~ des series ultra spherique par 1a m~thode des moyennes arithmetiques, Jour. de Math. pures et app1iques (9) 3 (1924), 107-187.

UNE APPLlCATION DE LA THEORIE DE L'APPROXlMATION A

L'ETUDE DES FONCTIONS HOLOMORPHES

* P.M. Gauthier

INTRODUCTION.

Recemment J. Clunie [3] a d~montr~ Ie resultat

remarquable qu'il existe des fonctions meromorphes

distinctes f et g dans Ie disque unite satisfai­

santes a

(1) Iim X(f(z),g(z» = 0 , Izl~I

ou X designe Ia distance sur Ia sphere de Riemann.

Depuis P. Lappan [5] et H. Allen [1] ont obtenu fonc­

tions holomorphes jouissantes des memes proprietes.

* Subventionne par Ie C.N.R. du Canada et par Ie Gouvernement du Quebec.

113

114 P.M. Gauthier

Dans cette note, en nous appuyant sur 1e th~oreme

dtArak~lian sur 1 tapproximation par fonctions ho10mor­

phes, nous donnons une demonstration extremement courte

de ces r~su1tats. Aussi nous a110ns r~pondre A une

autre question pos~ par M. C1unie 10rs d'une visite A

Montr~a1 en 1971.

Designons par D 1e disque unite dans 1e plan

* comp1exe, par D 1a compactification Dt A1exandrov,

par E un (relativement) ferme dans D, par C(E)

1es fonctions continues (A valeurs comp1exes) sur E

par CH(E) 1es fonctions continues sur E et ho10-

morphes Altinterieur EO de E, par H(D) 1es

fonctions ho10morphes dans D, et par H(D) 1es

1imites uniformes sur E de fonctions dans H(D) •

A10rs evidemment on a

1e ce1ebre theoreme d tArak€lian nous dit:

THEORtME A. (Voip 12]J. H(D) = CH(E) si et seuZement

* si D \E est connexe et ZocaZement connexe.

Approximation des Fonctions Holomorphes

1. COMPORTEMENT A LA FRONTIERE

Par des techniques astucieuses du genre Wiman-

Valiron, M. Clunie demontre:

THEOREME B. Il existe des fonations h et k

morphes dans D telles que

(2) lim Ih(z) I + Ik(z) I = 00

Izl~l-

De la, il deduit en quelques lignes:

THEOREME C. Si h et k sont les fonations du

Th~or~me B, alors les fonations

f = k/h et g = (k-l)/h

satisfont ~ (1).

holo-

115

Notre raccourcissement consiste a donner une courte

demonstration du Theoreme B. Pour cela nous avons

besoin d'un theoreme ayant un certain interet en lui­

m~me. Par un domaine spirale E nous entendons un

ensemble de D dont la frontiere (dans la topologie de

D) est une courbe simple a(t), _00 < t < +00 , satis­

faisant au conditions:

la(t) I ~ 1 et arg a(t) ~ +00 ,

lorsque Itl ~ 00 •

116 P.M. Gauthier

~ , THEOREME 1. Soit E un domaine spiraZe. AZors iZ

existe une fonation G 3 hoZomorphe dans D 3 teZZe que

(3) Re G(z) + +00, lorsque Izl + 1 dans E.

Ce theor~me entratne 1e Theor~me B. En fait, soit

EI et E2 deux domaines spira1es dont 1a reunion est

D. Soient G et k 1es fonctions correspondantes a EI et E2 respectivement d'apr~s 1e Theoreme 1, et

posons h = exp G. A10rs h et k satisfont a (2)

et en plus h n'a pas de zeros. Donc pour ce choix 1e

Theoreme C nous donne deux fonctions ho1omorphes

satisfaisantes a (1).

Nous remarquons que 1e Theoreme 1 genera1ise un

theor~me de W. Schneider [6] qui dit qu'i1 existe une

fonction G ho1omorphe et non-bornee dans D, mais

bornee dans D\E.

Demonstration du Theoreme 1. Soit F une repre­

sentation conforme de EO sur 1a demi-bande

Re w > 0, 11m wl < 1 ,

teIle que

F(z) + 00, lorsque Izl + 1 dans E.

Approximation des Fonctions Ho1omorphes

Par 1e Th~oreme A d'Arake1ian, i1 existe une

fonction G ho1omorphe dans D et te11e que

IG(z) - F(z) I < 1, z E E •

117

A10rs G satisfait ~ (3) et 1e Th€or~me 1 est d~montr~.

"-

2. UN PROBLEME DE CLUNIE

Nous rappe10ns un th€or~me de R. Remmert (non­

pub1i€) •

~ "-

THEOREME D. IZ existe trois fonctions g,h,k hoZo-

morphes dans D dont une injective, teZZes que

Ig(z) I + Ih(z) I + Ik(z) I ~ 00, Zorsque Izl ~ 1 .

En vertu des Theoremes B et D, M. C1unie posa 1e

probleme a savoir si on peut supposer h injective dans

1e Theoreme B. La reponse est non. En fait supposons

par contradiction que h est injective. A10rs si f

et g sont 1es fonctions du Theoreme C, f - g = l/h

est injective et donc ades 1imites angu1aires non­

nu11es presque partout sur 1e cercle unite (voir [4,

p. 56]). Soit p un tel point du cercle unite et

{z} une suite qui tend vers p dans un angle de n

Stolz. Puisque f et g satisfont a (1) i1 en suit

118 P.M. Gauthier

que si fest borne sur {z } n

a10rs f - g tend vers

zero sur {z} ce qui est exc1u. On en conc1ut que n

f a 1a limite angu1aire infinie en p et donc f a

limite angu1aire infinie presque partout ce qui est

absurde [4, p. 146].

, , REFERENCES

1. Allen, H.: Distinct ho10morphic functions with identica1 boundary va1ues (a paraitre).

2. Arake1ian, N. U.: Approximation comp1exe et proprietes des fonctions ana1ytiques. Actes, Congres intern. Math. , 1970 , Tome 2, 595-600.

3. C1unie, J.: On a problem of Gauthier, Mathematica 18 (1971), 126-139.

4. Co11ingwood, E.F. et Lohwater, A.J.: The theory of Cluster sets. Cambridge University Press, Cambridge, 1966, MR 38 #325.

5. Lappan, P.: A note on a problem of Gauthier. Mathematika 18 (1971), 274-275.

6. Schneider, W.J.: An e1ementary Proof and extension of an examp1e of Va1iron (a paraitre).

LINEARIZATION IN CERTAIN NONCONVEX MINIMIZATION

PROBLEMS AND GENERALIZED SPLINE PROJECTIONS

Joseph W. Jerome

1. INTRODUCTION.

The problem of minimizing the curvature, in the

L2 norm, of smooth functions f with square inte­

grable second derivatives, subject to certain inter­

polatory constraints on f, leads directly to the

consideration of constrained nonconvex minimization

problems in a Hilbert space and, more generally, in

a Banach space. In [6] and [7] an existence and

convergence theory was developed for the solutions of

extremal problems of the form

Research supported by National Science Foundation Grant GP-32116.

119

120 J.W. Jerome

(1.1) IITs 11 = min IITul1 'p UEU p

where T is a (possibly nonlinear) mapping of the

real Sobolev space wrn,P(a,b) into LP(a,b),

1 <p.::."" and U is a closed convex subset of

wrn,PCa,b) these spaces will have their usual meaning

throughout. In 16], a thorough analysis of the problem

of minimum curvature was made and a necessary condition

was given for solutions s in terms of the non­

negativity of the Gateaux derivative of the objective

functional at s on the cone {u: u + s E U} from

which the nonlinear differential equation satisfied

locally by solutions and the global smoothness

properties of solutions were deduced. Such results

are valid in the more general setting as we describe

in §2 of the sequel for 1 < p < "" and yield conditions

which are also sufficient if the objective functional

is convex as is the case, for example, when T is a

linear operator, a case treated by Golomb [3]. In §2,

we also discuss certain convex minimization problems

"" in Land the spline-type analytical character of

solutions. However, in §3, our results go beyond this

to include a characterization of Kuhn-Tucker type for

the solutions of classes of nonconvex minimization

Linearization and Spline Projections 121

problems, including problems of the form (1.1), in

Banach spaces. The nonnegative objective functionals

considered are of the form ~.~ where ~ is Gäteaux

differentiable and ~ is a seminorm on a suitable

space. Dur characterization theorem, which includes

both necessary and sufficient conditions for a

solution, makes fundamental use of results contained in

the book of Pshenichny; [11]. This analysis is carried

out in §3 and an application to problems of the form

(1.1) for 1 ~ p ~ 00 is made at the close of this

section in Corollary 3.3. This analysis is intended

as a preliminary step to the problem of obtaining an

algorithm for the determination of solutions of (1.1).

In §4, we consider the problem of constructing

generalized interpolating spline functions which are

obtained as projections via abilinear form

(1.2) B(u,v) =

on the Sobolev space wrn,2(a,b) , where

00

b _ 1 mm and

b .. E L (a,b) . No assumptions whatsoever are made 1J

concerning the nonnegativity of B(u,u) or the

symmetry of B(u,v). In fact, the spline solution

122 J.W. Jerome

may fail to minimize B(u,u) in any suitable way.

However, we obtain a characterization theorem, valid

in all cases, which leads to the interesting result

that any nonsingular linear differential Euler operator

A has associated interpolating spline functions,

locally in the null space of A. Previous results of

this type required either direct or indirect (e.g.

A = L*L) assumptions on the nonnegativity of B.

Essential use here is made of elegant results of Aubin

[1], particularly a generalized integration by parts

formula, which yields the characterization theorem.

Although the notion of a spline as a projection is

quite old, the consideration of general bilinear

forms as the context for such projections is quite

recent (cf. [8] and [10]). The present work should

be viewed simply as an extension of the work of [8]

and [10] (the former in the case of nonsingular

problems) in which the universal existence of genera­

lized spline functions is deduced for any bilinear

form of the type (1.2), thereby eliminating nonnega­

tivity assumptions. This result is contained in

Theorem 4.5 and may be considered independently of the

remainder of the paper. On the other hand, it is

related to the expansion, through quadratic terms, of

the nonconvex objective functionals mentioned earlier.

Linearization and Sp1ine Projections

The quadratic term in such express ions is in general

not globa11y nonnegative, as in the convex case.

Another interesting aspect of the projection approach

is that we also obtain a c1ass of sp1ines which is,

in a sense, dual to the usua1 c1ass, i.e., we obtain,

a1ternate1y, sp1ines satisfying inhomogeneous forced

boundary conditions and homogeneous natural boundary

conditions on the one hand and sp1ines satisfying

homogeneous forced boundary conditions and inhomo­

geneous natural boundary conditions on the other.

Such sp1ines are re1ated by so-ca11ed comp1ementarity

princip1es. MOre generally, we are ab1e to define

sp1ines which satisfy nonhomogeneous forced and

natural boundary conditions. In the particu1ar

123

examp1e of piecewise cubics, this means that the sign

of the jumps in the second derivative can be contro11ed

at noda1 points, thus producing 10ca11y convex or con­

cave interpo1ants, if desired. Theorem 4.6 contains

the precise statement of this in general setting.

Notice fina11y, that by producing sp1ines with

specified jumps in higher order derivatives, we can

direct1y construct Green's functions for multipoint

boundary va1ue problems.

124 J.W. Jerome

2. ANALYTICAL CHARACTERIZATIONS.

In this section we examine the structure of

solutions of certain minimization problems in

LP(a,b), 1 < P ~ 00. The results discussed here for

00

the L (a,b) case are expository in nature and are

quoted from [2]. An interesting consequence of these

results (cf. Theorem 2.5) is that the minimization

problem

= inf {IIDmf:f E tf1,oo(a,b):f(x.) = r. 1 1

o < i < n}

for prescribed a = x < ••• < x = b o n

and r , ... , r o n

has a unique spline solution of degree n (there may

be other solutions however). The results discussed

for LP(a,b), 1 < p < 00 , are new, however, and

enable one to deduce, in certain cases, that the

solutions of (1.1) satisfy locally a nonlinear

differential equation of the form (cf. Theorem 2.1)

(2.2)

where T is a nonlinear differential operator of

order m, DT(s;') is a linear differential operator

of order m which is the Gateaux derivative of T at

Linearization and Spline Projections

* sand DT(s;·) is its formal adjoint. For the

case T = Dm this result was obtained by Golomb [4].

We consider first the case of p for 1 < p < 00

Let T be a mapping of wm,P(a,b) into LP(a,b)

given by

(2.3) m-l m Tf = X ( • ,f ( • ) , .•. ,D f ( • ) ) D f

m-l + w(·,f(·), ... ,D f(·»

125

where X and ware continuous real-valued functions

on [a,b] x ~. We have the following

THEOREM 2.1. Let U be any alosed aonvex subset of

Wm,P(a,b). Let T: wm,P(a,b) + LP(a,b) be given by

(2.3). Then the minimization problem

(2.4) IITSlb = inf IITfl1 = Cl

fEU

has a solution s E U provided there is a bounded

minimizing sequenae in wm,P(a,b) , i.e., a bounded

sequenae {f} c U suah that IITf 11 + Cl. If, n n

moreover, the funations X(T,E; , ••• ,E; 1) o m- and

w(T,E; , ••• ,E; 1) have partial derivatives with respeat o m-

126 J.W. Jerome

to ~ , .•. ,~ 1 whiah are aontinuous on [a,b] x ~ o m-

then the mapping T has a Gateaux diffepential

DT(f ;.) at evepy f € wm,P(a,b) whiah is a o 0

bounded linear diffepential opepatop mapping wm,P(a,b)

into LP(a,b) given by

m-1 m + x(·,f (·), •.. ,D f (·»D f . o 0

The funational

has a GG.teaux diffepential D8 (f ;.) at every o

f € Wm,P(a,b) whiah is a aontinuous linear funational o

on wm,P(a,b) given by

(2.6) D8(f ;f) o

fop eaah f € wm,P(a,b) FinaZZy, if s is any

solution of the extpemal ppoblem (2.4), then

Linearization and Sp1ine Projections 127

ns(s;e) ~ 0 for aZZ e in the convex support cone

(2.7) K = {e e = A(f-s) for some A > 0 and some f € U}

i.e . .J b -1

(2.8) f ITsl P signum TS·nT(s;e) > 0 for atz e € K • a

ConverseZy.J if slip is convex.J as is true when T

is Zinear.J then (2.8) is sufficient for s to be a

soZution of (2.4).

PROOF. The existence resu1t is a consequence of

Theorem 2 of [7]. (2.5) is a consequence of the

multidimensional Tay10r theorem [3, p. 57]; indeed,

for each fixed T € [a,b] if we set

(2.9) s (~ , ... ,~ ) = X(T,~ , ... ,~ 1)~ Tom 0 m- m

+ W(T,~ , ... ,~ 1) , o m-

then we have

(2.10) s (~+tn , ••. ,~ +tn ) = s (~ , ... ,~ ) TO 0 m m TO m

m + o(t .I Inil) as t ~ 0

1.=0

128 J.W. Jerome

where the order expression is uniform in T E [a,b]

Thus, choosing and i n. = D f(T) . and 1

using (2.9) and (2.10), we deduce that DT(f •• ) 0'

exists as a bounded linear operator from wm,P(a,b)

into LP(a,b) given by (2.5). (2.6) results from an

application of the chain rule to the composition

mapping 8 = 11· IIP • T Now if s is any solution

(2.4) and e E K = K , then, for all sufficiently s

small A > 0 we have s + Ae E U and, hence,

Letting A tend to zero yields

D8(s;e) > 0 for all e E K .

The converse implication, when is convex,

follows from well-known results of convexity [11,

Theorem 2.1] if Ts I 0 and is trivial if Ts = 0 •

COROLLARY 2.2. Let T be a mapping from Wm,P(a,b)

into LP(a,b) given by (2.3) suah that X and w

have aontinuoUB partiaZ deriviatives with respeat to

~ , ... ,~ 1 on [a,b] x Rm • Let U be a nonempty o m-

of

Linearization and Sp1ine Projections

aonvex subset of wm,P(a,b) aonsisting of funations

f satisfying arbitp~ affine inequaZity aonstpaints

at nodaZ points a=x < ••• <x =b on linear o n

aombinations of depivatives of f thpough opdep

m - 1. If s is a soZution of the minimization

ppobZem (2.4), and, if the fopmaZ adJoint [DT(s;.)]*

of DT(s;o) exists as a Zinear diffepentiaZ opepatop

of opdep m with integpabZe aoeffiaients, and, if

(2.11) (m-1) x(·,s(o), ••• ,s (.» > C > 0

on [a,b], then ITsl P- 1signum Ts is in

m-1 C (xi ,xi +1) with (m-1)th dePivative absoZuteZy

aontinuous fop eaah i = 0,1, ••• ,n-1 and

(2.12) a. e. .

Mopeovep, aeptain highep opdep aontinuity aonditions

ape satisfied by ITsl P- 1signum Ts aaposs intepiop

nodaZ points x1 ' ... ,xn_1 ; speaifiaaZZy if s

satisfies affine HePmite-type aonstpaints of the foPm

129

j = 0, ... ,k.-1, ~

o < i < n ,

then

130

(2.14) j =

J.W. Jerome

m, ••. ,2m-k.-1 • 1

PROOF. The assumption (2.11) ins ures that DT(S;·) is

a nonsingular linear differential operator of order m

with continuous coefficients on Ia,b] . Now, if

and Xi+1 are adjacent nodes and if ep is an

infinite1y differentiab1e function with compact

support in (xi ,xi +1) then ep and -ep are in the

convex support cone K and it fo110ws from (2.8)

that

x i +1 -1 J {ITsl p signum Ts}DT(s;ep) = 0

xi

so that ITsl P- 1signum Ts is a solution, in the

sense of distributions, of

(2.15)

x . 1

* Since [DT(s;·)] is a nonsingular

linear differential operator of order m with

integrab1e coefficients, every distribution solution

is a c1assica1 solution in the sense described in the

statement of the theorem [5, Chapter 8]. The final

statement is a consequence of an argument given in

Linearization and Spline Projections 131

even more general form in 19, Chapter 7].

We remark that the Euler equation (2.12) for the

special case T = Dm was obtained in the case of

Lagrange-Hermite interpolation by Golomb [4] who also

obtained higher order continuity conditions across the

nodes and lower degree of the extremal solution at

infinity. Although he stated these only as necessary

conditions it is clear that they are also sufficient

since they lead to (2.8) for a convex minimization

problem.

In the remainder of this section we consider the

minimization of 11 Lfll, L a nonsingular linear p

differential operator of order m for 1< p ~oo ,

where f is subject to so-called extended Hermite­

Birkhoff constraints. Consider then a mesh

••• < x = b n

of [a,b] and, associated with each of the points xi'

consider the continuous linear functionals L.. on 1.J

wrn,P(a,b) defined by

(2.16) m-l

L .. f = I a~~)f(\))(x.) 1.J \) =0 1.J 1.

j

i=O, ... ,n,

132 J.W. Jerome

(v) for prescribed real numbers a.. such that, for each

~J

(0) (m-l) i , the k. m-tuples (a .. , .•. ,a. j ) are linearly ~ ~J ~

independent; here 1 < k < m for i- i = O, .•. ,n and,

at x and x , the derivatives are taken in the o n

limiting sense.

Let L be a nonsingular linear differential

operator of the form

(2.17) m-l .

L = Dm + l c.DJ j-o J

where c. E C[a,b] • Now let r .. , j J ~J

O, ••• ,k.-l, ~

i O, ... ,n be prescribed real numbers. Consider,

for 1 < P 2 00 , the minimization problem

(2.18) IILs I~ = a = inf {"Lf I~ fE~,P(a,b)

L .. f ~J

r .. ~J

j = O, ... ,k.-l, i = O, ... ,n} . ~

THEOREM 2.3. The minimization probtem (2.18) has a

sotution for 1 < P 2 00. Por fixed interpotation

vatues r .. the ctass S of sotutions is a convex ~J

Linearization and Sp1ine Projections

set. Among aU soZutions in S there exists a

soZution s* with the property that on each sub-

intervaZ ° ~ i ~ n-1 , the L P

norm of

is minimaZ in the foZZowing sense: Let S = S 1

133

and, for 2 < i < n , Zet S. consist of aZZ soZutions - ~

to the minimization probZem

SES. 1} ~-

then S n

is nonempty, i.e., there is an s* in S n

PROOF. For p = 00 , this is Theorem 1 of [2]; the

proof there, however, i8 valid for 1 < P < 00 also.

Now for each fixed i=O, ... ,n let A. ~

~x m matrix

A. = (a~~» ~ ~J

where j denotes row and v co1umn indices.

be the

Let Ä. ~

be any nonsingular m x m argmentation of Ai' Let

H. ~

be the inverse of the transpose of A

A. ~

134 J.W. Jerome

If operators o are defined on suitab1y smooth v

functions by

(2.19) 0 f = v

for v = O, .•• ,m-1 and if operators Rij

by

are defined

(2.20) j = 0, •.• , m-l , i = 0, ••• , n

then [cf. Lemma 3.1], if the notation

by

[.]. is defined 1

and

[~]. = ~(X.+) - ~(x.-), for 0 < i < n , 111

[ ~ ] = Hx +), [ ~ ] = Hx -) , o 0 n n

we have, for i = O,l, .•• ,n ,

(2.21) m-1 m-1 l njg(x.)[Ojf]. = l L.jgIR .. f]i

j=o 1 1 j=o 1 1J

b * = f [LfLg - L Lf·g]

a

Linearization and Sp1ine Projections 135

for all f,g for which (2.21) is meaningfu1. Notice

that A, ~

induces operators L .. ~J

for k < • < m-1 • i - J

m THEOREM 2.4. Suppose cj E C Ia,b] and 1 < p < 00 •

Then s is a solution of the minimization problem

(2.8) if and only if s E wrn,P(a,b) and

(i)

(2.22) (ii)

(iii)

on

i = 0, ... ,n-1 ,

L, ,s ~J

r, , ~J

j = 0, ... ,k.-1, i = O ... ,n ~

[R .. s]. = ° if j ~J 1

k. , ... ,m-1, ~

i = O, ••• ,n ,

PROOF. The direct imp1ications are a consequence of

(2.8), (2.12) and the integration by parts formu1a

(2.21). Converse1y, if (2.22 i, ii, iii) are satisfied,

then, using (2.21), we deduce that

b -1 J rlLsl P signum Ls]Le °

a

for all e such that L. ,e ~J

0, j O, ... ,k,-l, ~

136 J.W. Jerome

i = O, ... ,n. We conc1ude from Theorem 2.1 that s

solves (2.18).

Before stating the final theorem of this section,

we require a 1exicographic ordering of the Lij , i.e.,

i if N = 0 and N. L k i = 0, ... , n with 0 1. \)'

\)=0

N N , we define n

= L .. 1.J

° < j < k.-1, i = O, .•. ,n • - - 1.

We make the fo110wing assumption:

(I) N ~ m + 1 ; for each \) = 1, .•. ,N-m+1 the

functiona1s L, .•. ,L +m 1 are consistent with respect \) \)-

to the null space of L, i.e., for any prescribed

nmnbers r\), .•. , rv+m_1 there is a unique function u

in the null space of L satisfying L.u=r., 1. 1.

\) < i < v+m-1 . We now quote the fo110wing theorem

from [2] for p = 00

THOEREM 2.5. Suppose that (I) is satisfied and that

L* exists with continuous coefficients and has the

property that its nuZZ space is spanned by a

Linearization and Spline Projections

Tahebyaheff system. Then thepe is a fundamental,

intepval, J = [x ,x J with 0 ~ vI < v2 ~ n and vI v2

v2 l ki ~ m + I suah that any two sol,utions of (2.18)

:i:= vI

fop p=

ILs 1 = Cl

'00 agpee on J. Mopeovep, i f s € S then

a. e. on J, whepe S denotes the aonvex

sol,ution set of (2.18). If s* € S is ahosen as in

Theopem 2.3 then s* is unique in S. Mopeovep,

137

s* satisfies the ppopepty that ILs*LI is equival,ent

to a step funation on (x ,x) with disaontinuities o n

pestpiated to xl'.· .,xn_l and, on (xi ,xi +1),

i = 0,1, ••• ,n-l, Ls* is equival,ent to a step funation

with at most. m - 1 disaontinuities.

3. A THEOREM OF KUHN-TUKER TYPE.

Let X be a Banach space and let ,j, be a '1'0

(possib1y nonconvex) objective functional defined on

X. For i = -k, .•• ,~l and i = i, ••• ,l , let <Pi

be so-ca1led constraint functionals and let M c X

138 J.W. Jerome

be defined by

(3.1) M = {x E X ~i(x) < 0 for i < 0 and ~i(x) = 0

for i > O} •

We are interested in obtaining necessary and sufficient

conditions for x E M to be a solution of the problem: o

(3.2) = inf{~ (x) o

X E M} •

Now a functiona1 ~ on X is said to be

quasi-differentiabZe at a point x E X if there exists

a convex weak* c10sed subset M(x) c X' (the dual of

X) such that the directiona1 derivative

(3.3) ~(x) ae = 1im ~(X+te) - ~(x)

t-+O+ t

exists for each direction e E X and satisfies

(3.4) l.!!(x) = ae sup A(e) AEM(x)

The notion of quasi-differentiabi1ity is thus a

genera1ization of convexity; indeed, if ~ is convex

on X then ~ is quasi-differentiab1e on X and

Linearization and Spline Projections 139

M(x) may be taken to be the subdifferential of ~ at

x , i.e., the set of all A E X' satisfying

~(y) - u(x) ~ A(y - x) for all y E X

and in this case M(x) is bounded in X' as weIl as

convex and weak* closed and M(x) satisfies (3.4).

In the sequel, we shall be interested in quasi­

differentiable functionals on X such that, for

certain x E X, M(x) is bounded in the Banach space

X' . Indeed, we have

LEMMA 3.1. Suppose that X E M is fixed. o Let the

functionals ~i' I < i < l , be linear and, for

-k < i < 0 , suppose that the ~i are quasi-differen-

tiable on X with the corresponding sets M. (x ) 1. 0

bounded in X' •

convex functional

Moreover, suppose that there is a

h. such that 1.

(3.5) hi(e) > sup A(e) , e EX, AEM. (x )

1. 0

and that there is a privileged convex cone ~ such

that, for each e E ~ , and eaah sufficiently smaU

t > 0 ,

140

x o

x(t) = x + te E M • o

J.W. Jerome

is a solution to (3.2)3 there exist

numbers t. 3 not all zero3 and functionals l.

A. E Mi(x) such that l. 0

for a11 e E ~

and such that t i ~ 0 for i < 0 and ti~.(x) = 0 l. 0

for i < 0 .

PROOF. Simp1y combine Theorem 4.1, p. 83, the

Coro11ary of p. 88 and Lemma 4.2, p. 89 in reference

[11] .

We are interested in special choices of ~o as

remarked in the introduction. Thus, let Y be a

Banach space, ~ a seminorm on Y and let W be a " mapping from X into Y which is Gateaux differen-

tiab1e, i.e., for each x E X there exists a con­

tinuous linear mapping DW(x;·) from X into Y

such that, for each y EX,

w(x+ty) - W(x) = tDW(X;y) + o(t) , as t ~ 0 .

Linearization and Spline Projections

The nonnegative real-valued functional ~ = ~.t/J on o

X is quasi-differentiable 111, p. 69] and, if

IDt/J(xo ;·)], denotes the transpose of Dt/J(xo ;·) and

141

M(t/J(x » o denotes the subdifferential of ~ at t/J(x )

o

then the set

M (x ) = [Dt/J(x ;·)]'M(t/J(x » 000 0

satisfies (3.4) with x = x and, moreover, M (x ) 000

is convex, weak* closed and bounded in X' [11, p. 69],

the boundedness following since M(t/J(x» is bounded o

in Y' •

THEOREM 3.2. Let the (aonvex) set M in (3.1) be

defined by aonvex funationaZs ~i for -k ~ i ~ -1

and Zinear funationaZs for 1 ~ i ~ l. Let

~ = ~.t/J be a (quasi-differentiabZe) objeative o

funationaZ on X where t/J is a ch.teaux differentiabZe

mapping of X into a Banaah spaae Y and ~ is a

semi-norm on Y. If x € M is a soZution of the o

extremaZ probZem (3.2) then there exist numbers t i ,

not aZZ zero, and funationaZs Ai € Mi(Xo) suah that,

142 J.W. Jerome

(3.6)

for all e e: ~ ,

and suoh that t i ~ 0 for i ~ 0 and ti~i(xo) = 0

for i < o. Here ~ is the oonvex oone defined by

(3.7) ~ = {t(x-xo) : t > 0, x e: M} •

Conversely, if x e: M and (J.B) is satisfied for o

i < 0, t > 0, then x is a solution of (J.2) o 0

provided the ~teaux differential D$(x ;.) has the o

property that $(M) c D$(x ;.)~ + $(x) and o -N 0

provided

-1 (3.8) -.L tiAi(e) > 0

~=-k

for all e e: ~ •

PROOF. We first prove the converse. If (3.6) and

(3.8) hold, then, 1etting e e: ~ be arbitrary, we

have,

Linearization and Sp1ine Projections 143

-1 ~ (x +te) - ~o(xo) o 2. - l tiA.; (e) < t A (e) < t Um --.;;;..o---=o::...-. __ --.,;:;_~

i=-k • - 0 0 - 0 t~ t

~($(xo) + tD$(x je»~ - ~ (x ) + o(t) < t 1im 0 0 0 - 0 t

t~

~($(xo) + tD$(x je»~ - ~($(x » = t 1im 0 0

o t~ t

< t [~($(x ) + D$(x je»~ - ~($(x »] - 0 0 0 0

the 1atter inequa1ity a consequence of the convexity

of ~. The resu1t is now a consequence of the

hypotheses t > 0 and $(M) c D$(x ;·)K_ + $(x ) • o 0 -N 0

rhe necessity is simp1y a consequence of Lemma 3.1.

Indeed, the cone ~ defined by (3.7) satisfies

x(t) = x + te € M o

for all o ~ t 2. l/t'

if e = t'(x - xo) € ~. MOreover, if i = 0 , then

the functiona1 h can be defined by o

h (e) = o

sup A(e) A€M (x )

o 0

e € X ,

144 J.W. Jerome

and the convexity of h is a consequence of the o

fo11owing inequa1ities.

~ (x +t(ae1+ße2))-~ (x ) 1im 0 0 0 0

t~ t

(~(a$(x )+atD$(x ;e1)) 1 +~(~$(X )+ßt~$(X ;e2))-~ (x )+o(t)

< 1im 0 0 0 0

- t~ t

(a[~($(X )+tD$(x ;e1)-~ (x )] 1 o 000

+ß[~($(x )+tD$(x ;e2)-~ (x )] = 1im 0 0 0 0

t~ t

For -k ~ i ~ -1 , we define

and (3.5) is a consequence of (3.4), the convexity of

~. , and the increasing property of 1

Linearization and SpIine Projections

for t > o. This concIudes the proof.

We are now prepared to state a major consequence

of Theorem 3.2. Let X = Wm,P(a,b) for m > 1 and

1 ~ p ~ 00. Let T be a mapping of wrn,P(a,b) into

LP(a,b) of the form

(3.8') Tf = (m-I) m X(·,f(·), .•. ,f (.»n f

(m-I) + w(·,f(.), ... ,f (.».

We have

COROLLARY 3.3. Let T be a mapping of Wm,P(a,b)

145

into LP(a,b) given by (3.8~ for 1 ~ p ~ 00. Suppose

that X(T,~ , ... ,~ 1) and W(T,~ , ... ,~ 1) have o m- 0 m-

continuous partiaZ derivatives with respect to

~ , ... ,~ 1 on [a,b] x Rm and that convex func-o m-

tionaZs ~-k' ... '~-I and continuous affine functionaZs

~I' ... '~l are prescribed on Wm,P(a,b). Let U be

defined by

146 J.W. Jerome

U = {f € ~,P(a,b) ~i(f) ~ 0, -k ~ i ~ -1 and

~i(f) = 0, 1 ~ i ~ L} .

Then, if f € U satisfies o

(3.9) IITf P = inf IITf I1 , o P f€U 1>

therae exist numberas t i , not aZZ zerao, and functionaZs

Ai € M.(f) such that ~ 0

fora aZZ e in the convex supporat cone

{t (f - f ) : t > 0, f € U} • o

Herae, Mi(fo) fora -k ~ i ~ -1 is the subdifferaentiaZ

of ~. at f , and ~ 0

M (f ) = [DT(f ;·)]'M(T(f » 000 0

wherae M(T(fo» is the subdifferaentiaZ of 11· I~ at

T(fo) and DT{fo ;·) is the cateaux differaentiaZ of

T at f o . AZso, t i ~ 0 fora i < ° and

Linearization and Sp1ine Projections

PROOF. The Gateaux differentiabi1ity of T fo110ws

as in Theorem 2.1 and the coro11ary is a consequence

of Theorem 2.2.

147

We remark that Theorem 2.1 contains sufficient

conditions for (3.9) to have a solution for 1 < p < 00

For p = 00 , sufficient conditions are given in [7,

Theorem 4]. Fina11y, the necessity of Theorem 3.2

and Coro11ary 3.3 is tru1y meaningfu1 on1y when no

affine equa1ity constraints are present.

4. THE UNIVERSAL EXISTENCE OF GENERALIZED INTER­

POLATING SPLINE FUNCTIONS.

Fo110wing Aubin [1], we sha11 describe the abstract

Hi1bert space framework for our proeb1m and the resu1ts

in general form, before proceeding to the app1ication

of interest. To this end, let Hand V be Hi1bert

spaces and B(u,v) abilinear form on V such that

(i) I: V ~ H is continuous ,

(4.1) (ii) B(u,v) is continuous on V,

(iii) V is dense in H.

148 J.W. Jerome

Let W be a Hi1bert space and r a linear mapping

of V into W such that

(i) r is a continuous mapping of V onto W,

(4.2)

and (ii) the kerne1 V of r is dense in H. o

We have

LEMMA 4.1. Let (4.1) and (4.2) be satisfied. There

exists a Zinear operator A with domain DA C V

dense in H satisfying

(4.3) B(u,v) = (Au,v)H for aZZ u E DA' V E V o

A is the restriction of a continuous Zinear operator

A from V into the duaZ V' of V and DA is a o 0 II

HiZbert space under the graph norm

(4.4)

A is continuous as a Zinear operator from DA into

H. Moreover, there exists a continuous uniqueZy

detePmined Zinear operator ~ mapping DA into the

duaZ W' of W such that

Linearization and Spline Projections

(4.5) B(u,v) = (Au,v)H + (Qu,rv)

for all u € DA' v € V

where (.J.) represents the duality pairing on

W' x W •

PROOF. We give only an outline and refer the reader

to [1, Chapter 6] for complete details. For each ....

u € V we define Au € V' by o

(Au,V) = B(u,v) , for all v € V o

DA is defined to be the space of u € V such that ....

149

Au € H ,where H is he re identified as a dense linear

subspace of V' • o

Since the graph

{(u,Au) U € D}

is closed in V' x H it follows that DA is complete

under the norm induced by (4.4). Now the bilinear form

B(u,v) - (Au,v)H

is continuous on DA x V and there exists a continuous

linear operator B from DA J.

into V c V' • o

150

The transpose r' of

range of

r' is

r 1

V o

is c1osed.

since V o

r

J.W. Jerome

has c10sed range, since the

It fo11ows that the range of

is the kerne1 of r . If M 1

V o

is a continuous right inverse of r' mapping

onto W' ,then n = MB satisfies the Lemma.

Remark: By identifying W' with W it is possib1e

to assert the existence of a unique continuous operator

n from DA into W satisfying

(4.6) B(u,v) = (Au,v)H + (nu,rv)W

for all u E DA' v E V •

Our next resu1t describes the equiva1ence of a

variationa1 problem with that of a genera1ized boun­

dary va1ue problem. Let P be an orthogonal pro­

jection of W into itse1f and set Q = I - P. Let

M be any continuous right inverse of r. We have

LEMMA 4.2. Let (4.1) and (4.2) be satisfied and let

u denote the kernel of pr. Then the boundary o

value problem

Linearization and Sp1ine Projections

(4.7)

(i) Au = 0

(ii) Pfu = t 1

(iii) QQu = t 2

151

has a soZution u € ~ whe1'e t 1 € PW and t 2 € QW

if and onZy if the1'e exists

(4.8)

U € U satisfying o 0

fo1' atz V € U • o

In this aase~ u and u are 1'eZated by the equation o

(4.9)

PROOF. Suppose u € DA is a solution of (4.7). Then,

by (4.5), which we rewirte in the form,

(4.10) B(u,v) = (Au,v)H + (PQu,Pfv)W + (QQU,Qrv)W '

for v € V , it fo110ws that

solution of (4.8). Indeed,

u = u - Mt o 1 is a

Pfu = 0 so that ·0

U € U and (4.8) fo110ws from (4.7) and (4.10) (with o 0

u rep1aced by u). o Converse1y, if U € U satisfies o 0

152 J.W. Jerome

(4.8) then u given by (4.9) satisfies (4.7 ii) and

since V E U we have, o 0

B(u,v) = 0 = (O,v)H for all V E V o

Thus, u E DA and Au = O. Fina11y, to deduce

(4.7 iii) we use (4.8), (4.9) and (4.10) to conc1ude

that

1. e.,

(4.11)

Now Qf maps

for all v E U o

U onto QW so that (4.11) imp1ies o

that t 2 - Q~u = O. This comp1etes the proof of the

lemma.

This lemma, of course, does not guarantee the

existence of a solution of (4.7); it mere1y states

the simu1taneous existence of solutions of (4.7) and

(4.8). To obtain the existence of solutions we make

the fo110wing assumptions.

Linearization and Sp1ine Projections

(i) I: V ~ H is compact, and,

(ii) there exist positive constraints C

(4.12) and a such that B(u,u) + C(u,u)H ~

a(u,u)V for all u € V •

As previous1y, let u o

denote the kerne1 of pr

and let B be the unique1y determined continuous

linear operator from its dense domain DB C Uo which

is comp1ete under the usua1 graph norm, into H

satisfying

(4.13) B(u,v) = (Bu,v)H for all u € DB, v € Uo •

B is a restrietion of the operator A defined

ear1ier. By (4.12 ii) and the Lax-Mi1gram theorem

it fo11ows that B + CI maps DB onto Hand by

(4.12 i) the Riesz-Fredho1m-Schauder theory is

app1icab1e to B + CI + AI and hence to B + AI •

Specifica11y, we have that for real A, B + AI is

a continuous linear injective mapping of DB onto

H except when A be10ngs to a countab1e subset E

153

of R with no finite accumu1ation points. If A € E

the kerne1 of B + AI is a finite-dimensional subspace

154 J.W. Jerome

with dimension equal to the dimension of the kernel of

its transpose B' + Aland the range of B + Aland

B' + A I are closed. Finally, B' + AI is a con-

tinuous linear injective mapping of H' onto

and only if AlE •

D' B

if

Now if we identify H with H' and view B as

a closed linear operator with dense domain DB in H,

then the transpose mapping B' is an extension of the

* usual adjoint mapping B whose domain D * is B

defined as the set of all h € H such that there

* exists B h € H satisfying

(4.14) * (Bu,h)H = (u,B h)H for all u € DB •

* A mayaIso be characterized as the operator induced

by

(4.15) * B (u,v) = B(v,u)

in the same manner that A is defined by (4.5). * A

is not the adjoint of A • It is easily seen that the

kernel of B' + Al coincides with the kernel of B

* B +Al and is contained in DA* for all real A •

Linearization and Sp1ine Projections

We may thus app1y the Fredho1m-Riesz-Schauder theory

using the 1anguage of the adjoint operator. If w

is the operator such that

(4.16) * B (u, v) * (A u,v)H + (wu,rv)w for all

we have the fo11owing existence theorem. Reca11 that

A E E if and on1y if -A is an eigenva1ue of B •

155

LEMMA 4.3. Suppose that (4.1), (4.2) and (4.12) are

satisfied. Then if 0 I E , i.e., if the only solution

u E DA to the boundary va lue prob lem

(i) Au = 0

(4.17) (ii) Pru = 0

(iii) Qnu = 0

is the zero solution, then (4.7) has a solution

U E DA for every t 1 E PW and every t 2 E QW • If

o E E , then (4.7) has a solution if and only if

t 1 and t 2 satisfy the oompatibility oondition

* (4.18) (t2,Qrv)w = (t1 ,wPv)w for atz v E N

156

where

* of B

J.W. Jerome

* N = {u E DA * : pr u = Qw u = O} is the nutt space

PROOF. If 0' Ethen by the Riesz-Fredho1m-Schauder

theory there exists Uo E DA satisfying

(4.19) (Bu ,v) = l(v) for all v E U o 0

where lEW' is given by

(4.20)

Thus u is a solution of (4.8) and hence (4.7). The o

characterization of 0 E E given by a non-zero

solution of (4.17) is immediate from Lemma 4.2. If

o E E , then (4.18), via the integration by parts

*1 formu1a (4.16), imp1ies that l E N where l is

defined by (4.20). The Riesz-Fredho1m-Schauder theory

again yie1ds a solution u to (4.19) and hence to o

(4.7). The converse is simi1ar.

m >

We proceed now to the app1ications. For a fixed

1 , let ~,2(a,b) be the usua1 real Sobo1ev

Hi1bert space and let B(u,v) be the continuous

bi1inear form on Wm,2(a,b) given by

Linearization and Sp1ine Projections

(4.21) B(u,v) =

where bmm = 1 on (a,b) and bij € L~(a,b),

1 ~ i,j ~ m •

LEMMA 4.4. B(u,u) satisfies the aoepaiveness in­

equaZity (4.12 iiJ fop positive aonstants C and

with V = Wm,2(a,b) and H = L2(a,b) .

PROOF. We sha11 first estab1ish the inequa1ity, for

u € Wm,2(a,b) ,

where C is given by

2 C = max [(m max II bij II~) ,

o<i,j<m-l L

Now by (4.21),

157

158 J.W. Jerorne

Using the elernentary inequalities, valid for E > 0 ,

(4.23)

We have,

(4.24)

122 2(u,v) 2 ~ - -ll u ll 2 - Ilvll 2

L E L L

1 2 rn-I. 2 B(u,u) ~ (1 - -)11 Dmull 2 - ~" L b .DJuli 2

E L j=o rnJ L

rn-I . 2 rn-I i. - III ): birnD~j" 2 + ( 4 bijD u,D1u) 2

~=o L i,J=o L

and choosing E = 2 and using the inequality,

rn-I i _i 2 I( 2 bi·D u,~u) 21 ~ (rn

i,j=o J L

we obtain (4.22) from (4.24).

Now the coerciveness inequality (4.12 ii) follows

frorn (4.22) and the well-known interpolation in­

equality, valid for 0 < E < 1, and 0 < j < rn ,

Linearization and Spline Projections

(4.25)

This completes the proof of Lemma 4.4.

Let ro, ••• ,rJ be J + 1 linearly independent

continuous linear functionals on wm,2(a,b) • Let

B(u,v) be given by (4.21). Then, by Lemma 4.1, with

159

H the closure in 2 L (a,b) of the intersection of the

kerneis of ro " •• ,rJ , w= RJ+l and ff = (r of, ••• ,r I),

we deduce the existence of a closed linear operator

A 2 in L (a,b) and continuous linear functionals

J (4.26) B(u,v) = (Au,v) 2 + I niuriv

L i=o

for all u € DA and v € wm,2(a,b) • Now let

O 2 i l < i 2 < ••• < iN 2 J be a subset of N < J + 1

of the integers O, ••• ,J. Let P be an orthogonal

nJ+l projection in K defined by

160 J.W. Jerome

0 if i :I ~, k = 1, ••• ,N

(4.27) [P (r 0' ••• , r J) ] i = r i if i = i k

k

The fo11owing theorem is a direct consequence of Lemmas

4.2, 4.3 and 4.4 with P defined by (4.27) and rand

Q the Cartesian products of r i and Qi •

THEOREM 4.5. Let B(u,v) be a biZineaP fopm on

Wm,2(a,b) defined by (4.21) and Zet r1 , ••• ,rn be

n ZineaPZy independent aontinuous ZineaP funationaZs

on Wm,2(a,b). Let I = {i1 , .•• ,iN} be a nonempty

subset of 0, ... ,J . Let A and Q. satisfy (4.26). ~

Then, if the boundary vaZue ppobZem

(i) Au = 0

(4.28) (ii) r u = i 0 i I. I

(iU) Q u = i 0 i E I

has onZy the identiaaUy zepo soZution in DA' the

boundaPy vaZue ppobZem

Linearization and Sp1ine Projections

(i) Au = 0

(4.29) i € I

i /. I

has a sol,ution u € DA for arbitrary real, numbers

u satisfies the identity

(4.30) B(u,v) = L tiQiv for al,l, v such that il1

riv = 0, i € I.

If (4.28) has nontrivi~l, sol,utions, then (4.29) has a

sol,ution if and onl,y if to, ••• ,tJ satisfy the

compatibil,ity condition

(4.31)

for aU v in the nontrivial, l,inear space

Here wo, ••. ,wJ satisfy the genera1ized adjoint

integration by parts re1ationship

161

162 J.W. Jerome

* * J B (u,v) = (A u,v) + .I wiuriv 1.=0

* for all u E DA* and v E Wm,2(a,b), A derives from

* (4.16) and B (u,v) is defined by (4.15).

Our final theorem presents an app1ication of

Theorem 4.5 to the case where the operators r. are 1.

extended Hermite-Birkhoff functiona1s. Specifica11y,

we use the notation of §2 and we consider as given a

mesh a = x < ... < x = b and k. functiona1s Lij 0 n 1.

associated with each point x. of the form (2.16), 1.

where the corresponding matrix Ai of coefficients is

of fu11 rank. As in section two we augment Ai to

A A(\) obtain an invertib1e m x m matrix Ai = (aij ) • This

augmentation then yei1ds additional functiona1s L .. 1.J

at each xi ; more precise1y, we define ro, ••. ,rm(n+~l

as fo110ws:

(4.32) j = 0, ... ,m-1,

i = 0, ... ,n

The sub set I of the previous theorem is then

Linearization of Sp1ine Projections 163

defined by

(4.33) I = {O, •.• ,k -1,m, ••• ,m+k1-1, ... ,mn, ..• ,mn+k -1}. o n

Let H. = (hi(~» be the inverse of the transpose of ~ J

" Ai. Define, on suitab1y smooth functions, operators

o by v

(4.34) 0 = v

m-v-1 . . m I (_1)J+1nJ( I b DJ +v+i -m+1) i,j+v+1 •

j=o i=m-j-v-1

Here we have assumed that m oo bij E W ' (a,b) for all

i,j. Fina11y, as in (2.20) we define operators Rij •

Proper1y ordered, as ro, ••• ,rJ in (4.32)

(J = m(n+1)-1) , the operators [Rijf]i satisfy (2.21)

and, moreover, if we re1abe1 these operators no, ••• ,nJ

we have,

for all f,g for which the express ions are meaningfu1.

Here we have defined

164 J.W. Jerome

(4.36) A = m 'i' (_l)JDJ (b Di ) L iJ' i,j=o

The fo11owing theorem shows that the operator A of

(4.36) agrees with that of the previous theorem. By

the set W we mean

W o

o

THEOREM 4.6. Suppose that the coefficients bij of

B(u,v) are in ~,oo(a,b) and that k. extended ~

Hermite-Birkhoff functionals

i = O, ... ,n, are specified.

defined by (4.36) with domain

Lij at the points xi'

Then the operator A

n-l (4.37) D = Wm,2(a,b) n W2m,2(.u (xi,xi +1» n Wo

~=o

is a closed linear operator in

its domain, is characterized by

2 L (a,b) which, with

(4.38) (Af,g) 2 = B(f,g) for all f E DA' L

g E U o

Linearization of Spline Projections

where U is the intersection of the kernels of the o

operators Lij , The boundary value problem

165

(4,39)

(i) As = ° (ii) Lij = r ij j = O"",ki -1, i = O"",n

(iii) [R'jS]' = r .. 1. 1. 1.J

i=O",.,n,

has a solution for any specified numbers r" provided 1.J

the auxiliary boundary value problem (4.39) with each

r ij = ° has only the zero solution. In the event

that the auxiliary homogeneous problem has nontrivial

solutionB 3 let NA* denote the linear subspace of

DA* of solutions of the (purely) homogeneous adJoint

boundary value problem. Then (4.39) has a solution if

and only if the compatibility condition (4.31) holds

for all v E NA* .

PROOF. If (4.38) is satisfied for A and DA the

theory of distributions and the integration by parts

formu1a (4.35) imp1y that DA has the asserted form

(4.37). Converse1y, if DA satisfies (4.37), then

166 J.W. Jerome

(4.35) implies that (4.38) holds. The result is now

a consequence of Theorem 4.5 and earlier results.

b Remark: If B(u,u) = f (Lu)2 , where the coefficients

a m= of L are in W' (a,b) , it follows that (4.39) always

has a solution if the conditions in (iii) are homo­

geneous (cf. [9]).

REFERENCES

1. Aubin, J.P.: Approximation of Elliptic Boundary Value Problems, Wiley Interscience, New York, 1972.

2. Fisher, S.D. and Jerome, J.W.: Existence, charac­terization and essential uniqueness of solutions of = L minimization problems, Acta Mathematica (Sweden),

submitted.

3. Goffman, C.: Calculus of Several Variables, Harper and Row, New York, 1965.

4. Golomb, M.: Hm,P extensions by Hm,p splines, J. Approx. Th. 7 (1972), 238-275.

5. Halperin, I.: Theory of Distributions, University of Toronto Press, Toronto, 1952.

Linearization of Sp1ine Projeetions

6. Jerome, J.W.: Minimization problems and linear and nonlinear sp1ine funetions. I : Existenee, SIAM J. Numer. Anal., to appear.

7. Jerome, J.W.: Minimization problems and linear and nonlinear sp1ine funetions. 11 Convergenee, SIAM J. Numer. Anal. (submitted).

8. Jerome, J.W. and Pieree, J.: On sp1ine funetions determined by singular se1f-adjoint differential operators, J. Approx. Th. 5 (1972), 15-40.

167

9. Jerome, J.W. and Sehumaker, L.L.: On Lg-sp1ines, J. Approx. Th. 2 (1969), 29-49.

ID. Lueas, T.R.: M-sp1ines, J. Approx. Th. 5 (1972), 1-14.

11. Psheniehnyi, B.N.: Neeessary Conditions for an Extremum, Maree1 Dekker, Ine., New York, 1971. (Translation).

ON THE CONVERGENCE OF CUBIC INTERPOLATING SPLINES

Tom Lyche & Larry L. Schumaker

1. INTRODUCTION.

Given n > 2 , a partition /J. = {o = x < xl < n 0

< x = 1} of [0,1] , and a function n 0

C[O,l] g(O) = g(l)} f E C[O,l] = {g E , let P f n

be the periodic cubic sp1ine interpo1ating f at

(For a precise definition of P f , see §2). n

The fo11owing question has received considerab1e

attention recent1y (see e.g., [1 - 4, 6 - 9, 11 - 12]

and references therein): Given a sequence </J.) of n

partitions of [0,1] with

Research supported in part by AFOSR-69-1812-D.

169

170 T. Lyche & L.L. Schumaker

(1.1) 116 11 = n as n + 00 ,

what further conditions on (6) are needed to guaran­n

tee that the sequence of spIine interpoIants

converges uniformIy to f as n + oo? The study of

this question was stimuIated by the discovery by

Nord [9] of an exampIe of a sequence (6) satisfying n

o (1.1) and a function f E C[O,I] such that

IIf-Pflifo as n+ oo • n

Sufficient conditions on (6) n

to assure

11 f - P f JI + 0 as n + 00 have been given in terms of n

various measures of the spacing of the partition. One

such measure is

(1.2)

where h = i

In terms of

THEOREM A.

[0,1] with

m = n

xi - xi_I'

h max i/h O<i,~ <n j li-jl=I

i = 1, ••• ,n

m Marsden [7] proved n

If <6 ) is a sequenae n

m < m < 2 • 439+ , then n-

and h = h 0 n

o f parti tions of

Iim sup IIp 11 < 00

n n+oo ;

Convergence of Cubic Splines 171

o i.e. foT' evepy f € C[O,l].J IIf - Pnfll-+ 0 aB n -+ 00 •

Earlier vers ions of Theorem A were obtained by

MeirjSharma [8] (with m< 1:2), by CheneyjSchurer [3]

(with m < 2), and by Hall [6] (with m < 1 + 12). (See [7] for details of the history.)

The purpose of this paper is to establish analogs

of Theorem A for cubic natural spline interpolation

and for cubic type-I spline interpolation (see §2 for

the definitions). We relate our positive results to

certain negative results in §5. There are several

remarks in §6.

2. CUBIC SPLINES.

Given n > 2 and a partition tJ. = {O = x < x n 0 i

< < x = I} of [0,1], we denote the class of n

cubic splines with knots

(2.1)

tJ. by n

is a cubic polynomial

172 T. Lyche & L.L. Schumaker

We single out the fo11owing three subc1asses of S3:

(2.2) S3(6n ) = {s E S3(6n) : s(v)(O) = s(v)(l),

v = 0,1,2} ,

s"(O) = s"(l) = O} ,

These classes are the fami1iar periodic, natural, and

type I cubic splines.

(2.5) P f(x.) = f(x.) , i = O,l, ••• ,n n 1 1

Similar1y, given f E C[O,l] , we define N f E NS 3 (6 ) n n

(2.6) N f(xi ) = ° f(x.) = f(xi ) , i = O,l, ••• ,n n n 1

o 0

Pn defines a projection of C[O,l] onto S3(6n) ,

while N and 0 are projections of C[O,l] onto n n

Convergence of Cubic Sp1ines

In view of the e1ementary inequa1ities

and the we11-known fact that d(f,NS3 (ön» =

inf Ilf - sll and d(f,OS3(ön » converge to zero sENS 3 (ön )

for any f E C[O,l] provided ö satisfies (1.1), n

(e.g., this fo11ows from the resu1ts of [10] for

smooth functions or the results of [12]) it suffices

for us to s tudy "N 11 and "0 11. We dis cuss the n n

type-I sp1ines in §3 and the natural sp1ines in §4.

3. TYPE-I SPLINES.

The main resu1t of this section is

173

THEOREM 3.1. If <ö ) is a sequenae of partitions of n

[0,1] satisfying

174 T. Lyehe & L.L. Sehumaker

(3.1) m < m < 2.439+ for all n, n-

then 1im sup 11 0 11 < 00. Thus for any sequenae of n

n~

partitions \Ö > of [0,1] satisfying (3.1) and for n

any fEe [0 ,1], 11 f - 0 fll + ° aB n + 00 • n

Our proof of this resu1t is simi1ar to that used

by Marsden [7] for Theorem A. First we need a loea1

3 Let M(x,y) = (y - x)+ and let

M[X;~l' ••• '~r] denote the r - Ith divided differenee

of M as a funetion of y taken over ~1 < ~2 <

< ~r· Setting x_i =-xi and xn+i = 2 - xn_i '

i = 1,2,3 we eonsider the usua1 norma1ized B-sp1ines

(3.2)

We reea11 ° ~ NMi(x) ~ 1

X E [0,1] . We define

i = -1, ••. ,n+1 .

n+1 and L NMi(X) - 1

i=-l for

Convergence of Cubic Sp1ines

NMi(X) , i = 0,2,3, .•• ,n-2,n

(3.3) ~i(x) = NM1 (x) + NM_1(x), i = 1 ,

NMn_1 (X) + NMn+1 (x) , i = n-l •

By the construction of the ~. 's it is not difficu1t 1.

175

to see that they form a basis for OS3(ßn ) (e.g., the

symmetry of the knots guarantees ~'(O) = ~'(O) = o 1

~' (1) = ~'(1) = 0). Moreover, ° < ~ (x) < 2, n-1 n - i -

n i O,l, ••• ,n and L ~i(x) = 1 + NM_1 (X) + NMn+1 (x) ~ 3.

i=o

The estimate for 110 11 is based on the fol10wing n

lemma.

LEMMA 3.2. Then

n = 3 max I IA .. I

O,$j~n i=o 1.J

PROOF. For i = O,l, •.• ,n let si(x) E OS3(ßn ) be

the sp1ines satisfying si(xj ) = 0ij' i,j = O,l, ••• ,n •

176 T. Lyche & L.L. Schumaker

n n Then 0 f(x) = L f(xi)si(x) and 11 On 11 ~ L Isi(x)1 •

n i=o i=o

n Suppose ß = (ßij)i,j=o is such that si(x) =

n .L ßij~.(x). Then J=o J

But I = Aß and the resu1t f011ows.

PROOF OF THEOREM 3.1: The matrix A is exp1icitly

b c 0 0 0 0

(al + c_l ) bl cl

0 a2 b2 c2

A= an_2 b n-2 c n-2

0 0

0 an_1 b n-l (cn_l + an+l )

0 0 a b n n

Convergence of Cubic Splines

where for i = -l, ••• ,n+l

2 h i +2

ci = NMi (xi +1) = ~(h:-i-+-l~+-:h~i-+-2-=-) -::(h-i"':;;"'-'+-:h:-i-+-l-+~h-i-+-2-=-) ,

To estimate IIA-lill we follow Marsden [7] and use

(3.4)

where D is any matrix. We take D to be the

diagonal matrix with entries It is shown

2 in [7] that b-l < (m + m + l)(m + 2) if ß

i = (2m + 1) (m + 1) n

satisfies (3.1). Now

177

178 T. Lyche & L.L. Schumaker

DA - I ""

0 c /b 0 0 o 0

(a1+c_1)/b1 O cl/bI 0

a2/b2 0 c2/b2

a /b n-2 n-2 0 c /b n-2 n-2 0

o 0 a /b 0 n n

It is shown in [7] that ci/bi + ai +2/b i +2 < 1 for

i = O, ••• ,n-2 under the assumption (3.1). It remains

to show that (al + c_1)/b1 < 1 and (cn_1 + an+1)/

b n-1 < 1 . It suffices to consider the first of these.

By the symmetry in the choice of knots x_3 , x_2 ,

x_I' a1 = c_1 , so we need on1y show 2a1 < b1 . It

is an exercise in ca1cu1us to see that the maximum of

occurs for and (In

doing this, reca11 ho = h1 .) For this choice,

Convergence of Cubic Splines 179

2 2a /b < (m + ~)(m + m + 1) < 1 for m satisfying

1 1 - 4m l + Sm + 3

(3.1) .

The spline 0 f given above is not quite the n

usual type I-interpolating spline of the literature

(see e.g., [10]). Given f with f'(O) and f'(I)

defined, the type I interpolating cubic spline 0 f n

is defined by

(0 f)'(x.) = f'(x.), i = O,n • n 1. 1.

THEOREM 3.3. Let

f'(I) are defined.

f E C[O,I]

If (& > n

and suppose f' (0)

is a sequence of

parti tions wi th m < m < 2.439+ , then n-

lim 11 0 f - f 11 = ° . n n400

and

PROOF. Given f, choose a cubic polynomial p such

that p' (0) = f' (0) , p' (1) = f' (1) . For functions

g with g'(O) = g'(I) = 0, ° g = 0 g. Thus n n

180 T. Lyche & L.L. Schumaker

Ilonf - fll ~ IIOn(f - p) - (f - p) 11 + lIonP - pli

= 11 0 (f - p) - f - p) 11 + 11 ° p - p 11 • n n

The first term on the right converges to 0 as n + 00

by Theorem 3.1 whi1e the second converges to 0 by

weIl known resut1s for smooth functions (see e.g. [10]).

4. CUBIC NATURAL SPLINES.

In this section we prove

THEOREM 4.1. If (ßn> is a sequenae of partitions of

[0,1] with

(4.1) m < (1 + 1:13)/2 = 2.30+ and m < m for aU n-

then lim sup 11 N 11 < 00. Thus for any sequenae of n n~

partitions satisfying (4.1) and any f € C[O,l],

11 f - N fll + 0 aB n + 00 • n

n ,

The basic out1ine of the proof is the same as in

§3. First we need a basis for NS3 (ßn). With M(x,y)

as in §3 we let

Convergence of Cubic Sp1ines 181

(xi +2 - xi_2)Mlx;xi_2,···,xi+2]'

(4.2) ~i(x) = i = 2, ••. ,n-2

The

Mlx;x 3'x 2'x l'x], i = n-1 n- n- n- n

M[x;x 2'x l'x ]/(h 1 + 2h ), i = n . n- n- n n- n

{~}n is (except for a norma1ization) the basis i 0

constructed by Grevi11e [5] for NS 3(6n ). It is easi1y

n-2 verified that 0 < ~.(x) < 1,

- 1 -I ~i(x) ~ 1 , and

i=2

~ (x ) = ~l(x ) = ~ 1(x) = ~ (x ) = 1 • o 0 0 n- n n n

Before proving Theorem 2.1 we need a lemma whose

proof is on1y a slight variant of that used for Lemma

3.2.

LEMMA 4.2. Then

182 T. Lyche & L.L. Schumaker

PROOF OF THEOREM 4.1. The matrix A in Lemma 4. 2 has

the following form

o o A =

c n-2 o

where the Ci are as in §3 and

2 -1 2h )-1 . ~ (x 1) = h l(h 1 + h) (h 1 + ~n n- n- n- n n- n

We estimate /lA-1 111 by the formula (3.4), where

Convergence of Cubic Sp1ines

now we choose

a i = 1

ß , i = 2

Dii = 1 i = 3, ••• ,n-1

y , i = n

15 i = n+1 •

with a, 15 > 1 and 0 < ß, y < 1 to be chosen

1ater. Then

0 a2 b2_1 c2

0 0 b -1 3

DA - I =

a b -1 n-2 n-2 c n-2

0 ya 1 y-yc -1 n- n-2

0

y

0 0 öep (x 1) 15-1 n n-

183

Marsden showed (by e1ementary Ca1cu1us in a pre1iminary

version of [7]) that ci + ai +2 < bi +1 , i = 1,2, ••• ,11-3 ,

184 T. Lyche & L.L. Schumaker

under the assumption (4.1). It remains to show the

ll-norms of the first and last two columns do not

exceed 1. We concentrate on the first two, and

choose a,ß such that the norms are 1 - E, E > 0

sma11. Thus we require

(4.3)

The solution of this system is

a = 2-3a2+Ea2-2E

(1-a2+~o(x1» , ß =

a2+E+2~o(xl)-E~o(x1)

(1-a2+~o(xl»

For 1 < a and 0 < ß < 1 we need

(4.4)

The minimum of 1 - a2 + ~o(xl) and the maximum of

2a2 + ~o(xl) both occur for partitions with

hi +1 = hi/m, i = 1,2. In this case

Convergence of Cubic Sp1ines 185

3 m + 1

(m2+m+1) (m+1) (m+1) (2m+1)

It is easi1y checked that if (4.1) ho1ds then (4.4)

does too (if we take E sufficient1y sma11).

5. NEGATIVE RESULTS.

There is a definite limit to the extent to which

the ear1ier resu1ts can be improved.

THEOREM 5.1. For eaah fixed m> (3 + 15)/2 there.

exists a sequenae of partitions (ÄJ of [0,1]

satisfying (1.1) and mn ~ m, aZZ n, suah that

1im sup 11 p 11 = W'<'" n

= 00 •

By the uniform boundedness princip1e, Theorem 5.1

asserts that if m> (3 + 15) /2 there exists a sequence

of partitions with mn ~ m and IIÄnll + ° and a func-o

tion f € C[O,l] such the P f does not converge n

186 T. Lyche & L.L. Schumaker

uniforrnly to f. Similarly, there exist continuous

functions and partitions with m < m n- and

such that N fand 0 f do not converge to the n n

functions.

Aversion of Theorem 5.1 without the assertion

that "L\ 11 -+ 0 as n -+ 00 was proved by Marsden [7]. n

Without this property the theorem would not be a true

negative convergence result. The proof of Theorem 5.1

re lies on sign regularity properties of the matrix A,

and is deferred to a later paper.

6. REMARKS.

1. The periodic cubic spline has received consider­

ably more attention than the natural and type-I

splines. There are, of course, general results which

apply for smoother functions fand there are results

for continuous f when the partitions are quasi­

uniform (see e.g. [10, 12]). Hall [6] obtained a

version of Theorem 3.1 under the hypothesis m <m<oo n-

coupled with an additional hypothesis on another

measure of partition behavior. Cheney (unpublished

lecture notes) proved aversion of Theorem 4.1 assuming

Convergence of Cubic Splines

m < m < 2 (by entirely different methods). n-

187

2. The methods employed here can be applied in other

cases as suggested by Marsden. For example, they can

be used to show that for equally spaced knots periodic,

natural, or type-I splines of any order converge as

n + 00 for arbitrary continuous functions. (This is

of course also a direct consequence of the results of

Swartz/Varga [12].) However, for higher-order splines

it is increasingly difficult to estimate IIA-l"l. A

useful tool for this purpose is a lemma of de Boor [4]

which requires some sign-regularity properties of A.

Such sign regularity properties of spline bases can be

derived conveniently from the total-positivity of the

B-splines; one need only study the properties of

certain transformation matrices relating the desired

basis functions to the B-splines. We hope to make

further application of these ideas to higher-order

spline convergence.

3. The gap between the constants in the positive

results (2.439+ in Theorems A and 3.1 and 2.30+

in Theorem 4.1) and the constant (3 + 15)/2 = 2.62-

in Theorem 5.1 is annoying. There is hope that the

methods used here could be employed to close the gap.

188 T. Lyche & L.L. Schumaker

REFERENCES

1. Birkhoff, G.D. and de Boor, C.: Error bounds for sp1ine interpolation, J. Math. Mech. 13 (1964), 827-835.

2. Cheney, E.W. and Schurer, F.: A note on the operators arising in sp1ine approximation, J. Approx. Th. 1 (1968), 94-102.

3. Cheney, E.W. and Schurer, F.: Convergence of Cubic sp1ine interpo1ants, J. Approx. Th. 3 (1970), 114-116.

4. de Boor, C.: On the convergence of odd-degree sp1ine interpolation, J. Approx. Th. 1 (1968), 452-563.

5. Grevi11e, T.N.E.: Introduction to sp1ine functions, in Theory and App1ication of Sp1ine Functions, Academic Press, New York (1969), 1-35.

6. Hall, C.A.: Uniform convergence of cubic sp1ine interpo1ants, J. Approx. Th., to appear.

7. Marsden, M.: Cubic sp1ine interpolation of continuous functions, J. Approx. Th., to appear.

8. Meir, A. and Sharma, A.: On uniform approximation by cubic sp1ines, J. Approx. Th. 2 (1969), 270-274.

9. Nord, S.: Approximation properties of the sp1ine fit, BIT 7 (1967), 132-144.

10. Schultz, M. and Varga, R.S.: L-sp1ines, Numer. Math. 10 (1967), 345-369.

Convergence of Cubic Sp1ines 189

11. Sharma, A. and Meir, A.: Degree of approximation of sp1ine interpolation, J. Math. Mech. 15 (1966), 759-768.

12. Swartz, B.K. and Varga, R.S.: Error bounds for sp1ine and L-sp1ine interpolation, to appear.

191

AVERAGING INTERPOLATION

T.S. Motzkin & A. Sharma & E.G. Straus

1. INTRODUCTION.

Given a set X of N(~ n) real numbers there

exists a unique polynomial P of degree n - 1 n-l,f,X

or less that approximates best on X to a given real

function, that is, that minimizes the deviation o(f,P)

defined by the th power metric (1 < p < 00) with p

positive weights or (p = 00) by max If - pi on X

and there are at least n + 1 points on X , where

the difference f - P takes alternatingly positive

and negative values. When N = n the polynomial

P is the polynomial of interpolation, and for n-l,f,X

N = n + 1 and p = 00 we obtain the next-to-inter-

Research supported in part by NSF Grant GP 28696.

192 T.S. MOtzkin & A. Sharma & E.G. Straus

po1atory polynomial.

In arecent paper [3] we have estab1ished a

relation between interpo1atory and next-to-interpo1atory

polynomials on a finite set X and extended it to sets

X of multiple points. We have shown that for a given

p, 1 < p ~ m and N = n + 1 every positive1y

weighted mean of interpo1ators is a next-to-interpo1ator,

Le., it minimizes eS for a suitab1e choice of weights,

and in order to bring out this resu1t in its genera1ity,

we have introduced the concept of uniso1vence relative

to a given set of functiona1s on an-parameter fami1y

on n + 1 points. The object of the present paper is

twofo1d. First in §2 and §3, we extend the concept of

interpolation by considering "average interpo1ators"

that share with the given function certain va1ue

averages, instead of the va1ues themse1ves. Second1y

we study ac1ose1y connected minimization problem. This

enab1es us to genera1ize resu1ts of [4].

In §2 we formu1ate the general interpolation

problem. Our principa1 resu1t is proved in §3 and a

special case is discussed in §4. The trigonometric

ana10gue of Theorem 2 is formu1ated in §5. The special

case when the points of X are such that

ßXi + Y for all i is treated in some detail in §6.

Averaging Interpolation 193

In this ease we show how to obtain the polynomial of

interpolation explieitly. In §7 we generalize the

eoneept of relative unisolvenee introdueed earlier in

13]. §8 and §9 deal with possible applieations of

our prineipal results to extend the results of Motzkin

and Walsh [4].

2. A GENERAL INTERPOLATION PROBLEM.

Let X = (xl< ..• < xn) be n given real

numbers and let m be a given integer, 0 < m < n-l •

Let a. > 0, r = 1, .•. ,m. Set r

(2.1) A(z)

and

A (z) r

m = II(l+a. z)

1 r

= A(z)/(l+a. z) = r

m-l

a o

l a zll o ll,r

If l = (ll, ••• ,l ) are integers with n-m

1 .

o < l. < n-m-l , eonsider the funetionals Ll, •.• ,L - J - n-m

given by

194 T.S. Motzkin & A. Sharma & E.G. Straus

(2.2) m (I. )

L. (f) = I a f J (x +.) J p=o P P J

j = 1, ..• , n-m •

The problem is to find a po1ynomia1 p(x) of degree

< n-m-1 which satisfies

(2.3) L.(P) = L.(f) J J

j = 1, ... ,n-m •

We note the fo11owing special cases:

1. m = 0, I = (0, ••• ,0) . The po1ynomia1 P is

the Lagrange interpolation po1ynomia1 of degree

< n-1 •

2. m = 0, I = (0,1, .•• ,n-1). The po1ynomia1 P

is the Abe1-Gontcharoff po1ynomia1 of interpolation.

3. m = 1, I = (0, ... ,0) The po1ynomia1 P is

the next-to-interpo1atory po1ynomia1 of degree < n-2 •

If A(x) = 1 + u1x, u1 > ° then the po1ynomia1 p(x)

minimizes

j-1 1 I max u1 O. '-1 J J- , •.. ,n

where O. = P(x.) - f(x.) . J J J

4. m = n-1, I 1 = 0. Then the po1ynomial P is a

constant C given by

Averaging Interpolation

C = l a f +1 La, n-l /n-l

o ~ ~ 0 ~

where, as in the sequel, f j = f(xj ) .

The functionals (2.2) can be considered in terms

of operators on the n-vectors of functional values

f = (f1 , ... ,fn) , composed of the following three

basic operators:

195

(i) The differentiation operator nf = (fi, ••• ,f~) •

(ii) The truncating identity operator

Jf = (f1 , ... ,fn_1) .

(iii) The shift operator

Then if

operator

given by

(2.4)

A(z) is the polynomial (2.1), we have the

L = (L1 , ... ,L ) with n-m

Lf = (L1 (f), ••• ,L (f» n-m

196 T.S. MOtzkin & A. Sharma & E.G. Straus

which maps n-vectors into (n-m)-vectors. With on1y

a slight abuse of 1anguage, we can write

A(E) = (J+a1E) ••• (J+a E) = (J+a E)A (E) m r r

The conditions (2.3) can now be written as

(2.5)

where P = (P1 , .•• ,Pn) and L is given by (2.4).

3. TRE AX-POLYNOMIAL.

We sha11 restriet ourse1ves to the case when all

l. are zero. p(x) is then ca11ed an AX-po1ynomia1. J

First we prove

THEOREM 1. If A(z) of (2.1) has onZy negative ze~o,

then the~e exists a unique AX-poZynomiaZ p(x) of

deg~ee ~ n-m-1 , that is, a poZynomiaZ such that

m (3.1) La P(x +.)

0]..1 ]..I J

m

= La f +j o ]..I ]..I j = 1, ... , n-m .

Averaging Interpolation 197

For m > 1 and arbitrary r = 1, ... ,m, P(x) is the

poZynomiaZ which minimizes

(3.2)

. 1 m-1 max

j=l, .•. ,n-m-1 aJ - I \ a 0 I

r ~ ll,r ll+j

o = Q(x ) - f v v v v=l, •.. ,n

over aZZ poZynomiaZs Q(x) 01 degree < n-m-1 •

The po1ynomia1 P(x) satisfying (3.1) is thus a

solution of m (in general different) minimization

problems .

PROOF: In order to prove that (3.1) has a unique

solution, it is enough to consider f = 0, v

v = 1, ... ,n and show that this entai1s p(x) _ 0 .

The equations (3.1) can then be written as

(3.3) m

A(E)P:: Ir (J+a.E)P = 0, P = (P1 , ... ,Pn) . j=l J

Since a. > 0, j = 1, .•• ,m , it is easy to see that J

the operator n-vectors into

(n-1)-vectors is variation diminishing. After m - 1

steps, we arrive at the (n-m-1)-vector

198 T.S. Motzkin & A. Sharma & E.G. Straus

given by A (E)P . r

Since

(3.4) (J+a E)R = (J+a E)A (E)P = A(E)P = 0 r r r

it follows that if Rn-m+l ~ 0 , then veR) , the

-+ number of strong sign changes in R is n - m and so

v(p) ~ n-m , which is impossible for a polynomial of

are zero. degree ~ n-m-l. This proves that all Rj

Repeating the argument m times, we see that all Pj

are zero, i.e., P(x) = 0

give

If the f are not all zero, the conditions (3.1) v

which is equivalent to

o +a 0 = 0 +a 0 = ••• =0 +a 0 = 0 r,l r r,2 r,2 r r,3 r,n-m r r,n-m+l

where & = (0 1,·.·,0 m+l) = A (E)!. That is r r, r,n- r

n-m o I = -a 0 2 = •.. = +(-a) 0 m+l· r, r r, r r,n-

This proves that P(x) minimizes (3.2).

Averaging Interpolation

Remark: In order to find an exp1icit expression for

P(x) , we set

m

gj = La f +j 0 1111

j = 1, ••. ,n-m

and first solve the linear system of equations:

(3.5) m

La P +" 0 1111 J

g j , j = 1,..., n-m •

We introduce m parameters gn-m+1, ••• ,gn by the

equations:

(3.6) n-j L a11P 11+j = gj , j = n-m+1, •.• ,n • o

The two systems (3.5) and (3.6) in the n unknowns

P1 ",.,Pn can be solved exp1icit1y if we set

(3.7) b = 1 • o

It fo11ows by e1ementary computation that

(3.8) P = \I

\I 1, ... n •

199

200 T.S. Motzkin & A. Sharma & E.G. Straus

Since P(x) is a po1ynomia1 of degree ~ n-m-1 , the

divided differences of order n - m of the numbers

If we set w (x) = (x-x ) v v

(x-x +n ) , we can write these conditions as m v -m

equations:

(3.9) n-m PV+k l = 0, v = 1, ... ,m • k=o w~(xv+k)

Using (3.8), we get from (3.9) after some simp1ifi­

cation

(3.10)

with

c P,v =

n-v ~ c g = 0 v = 1, .•. ,m L. p v ll+V '

p=o '

ntm bp_k L. w' (x )'

k=o v v+k

~ b}l-k L. w'(x )' o v v+k

p = n-m+1, ••• ,n-v

1.1 = 0,1, ••• ,n-m

Solving (3.10) by Cramer's ru1e for g m+1,···,g n- n

and using (3.8) we determine all the P v and then

P(x) is determined by Lagrange interpolation.

Averaging Interpolation 201

Theorem 1 can be further genera1ized.

* * THEOREM 2. If A1(z), ... ,A (z) aPe n - m poZy-n-m

nomiaZs of degree ~ m eaoh having onZy negative zeros~

then there exists a unique AX-poZynomiaZ P(x) of

degree < n-m-1 ~ that is a poZynomiaZ suoh that

(3.11)

where

m * la jP(x +.) oll llJ

m * = la .f +j

o llJ II

* m * A. (z) = la .zp J 0 pJ

j = 1, ... ,n-m

For m ~ 1 ~ and aPbitrary r = 1, ... ,m~ P(x) is the

poZynomiaZ whioh minimizes

(3.12)

(3.13)

m-1 (j) max w./la o+j/' _ J pr P j-1, ... ,n-m+1 0

m-1 w. = II Cl

J 1 rll w = 1

o

(v = 1, ... ,n)

over aZZ poZynomiaZs Q(x) of degree < n-m+1 ~ where

202 T.S. MOtzkin & A. Sharma & E.G. Straus

m = II(l+o. ,z) =

1 rJ

The proof of Theorem 2 fo11ows the same 1ines as

that of Theorem 1 and is omitted.

Note that the weights wj in (3.12) are com­

p1ete1y arbitrary positive weights, thereby removing

the rather artificia1 restrietion to weights of the , 1

form o.J - in (3.2).

4. THE CASE m = 2 .

The case m = 2 is of sufficient interest and

i11ustrates Theorem 1 fair1y we11. Suppose A(z) = (1+o.1z) (1+a.2z), 0.1 ,0.2 > 0 and we seek to find the

minimum of

(4.1) j-1 1 I max 0.1 0, + o. 2oJ'+1 '-1 1 J J- , ..• ,n-

where 0v = Q(xv) - f v ' v = 1, ••• ,n over all

polynomials Q(x) of degree < n-3. If this minimum

A is taken by P(x) , then

Averaging Interpolation 203

(4.2)

. Pn- l + a 2Pn = f n_l + a 2f n + (_1)n-2A/a~-2

This system of equations can be easily solved. Indeed,

we have for v = l, ••• ,n-l

(4.3)

P = (-a )n-v(p -f ) + f v 2 n n v

1- r.a2)n-V

(-1) v-I laI + v-I

a l

v-I = (-a2)n-v(p -f ) + f + (-1) (n-v)A

n n v v-I a 2

Then by Lagrange interpolation

p(x) = ~ w(x) P LI (x-x )w'(x) v

v v w(x)

n = L(x-x.)

1 J

204 T.S. Motzkin & A. Sharma & E.G. Straus

Since p(x) is a polynomial of degree ~ n-3 , the

coefficients of n-l n-2 must vanish. This x x

requirement yields

n p n x P (4.4) I w' (~ ) = 0 , I v v o . w' (x ) =

1 v 1 v

From (4.3) and (4.4) we then have for a1 ~ a2 :

whence we have

n x f = \ v v

- l. w' (x ) 1 v

where D1 , D2 are determinants given by

Averaging Interpolation

n f n (-a )n-v

I w' (~ ) I 2 w' (x ) 1 v 1 v

Dl = x f n-v n n (-a2) Xv I v v I 1 w f(X) w' (x ) 1 v

n (-a )n-v n {-a )n-v

I 1 I 2 w' (x ) w'(x) 1 v 1

D2 = n-v n-v n (-al) Xv n (-(2) Xv

I w'(x) I w' (x ) 1 1 v

5. TRIGONOMETRIe POLYNOMIALS •

Theorem 2 has a trigonometrie analogue. We

suppose for this purpose that X has 2n + m + 1

points, {O ~ xl < ••• < x2n+m+l < 2~}. Suppose

205

206 T.S. Motzkin & A. Sharma & E.G. Straus

(5.1) * A. (z) J

m * = La .z].l = o 11J

m TI (1 +a. . z)

r=l rJ j = 1, •.. , 2n+ 1

* (5.2) A • (z) = rJ

=

Then the argument used in the proof of Theorem 2 can

give simi1ar1y

* THEOREM 3. If A.(z), (j = 1, .•. ,2n+1) are the J

polynomials given by (5.1), then there is a unique

AX-trigonomitrie polynomial T(x) of order n, such

that

(5.3) m * La .T(x +.) = o].lJ ].l J

m * La .f +. o ].lJ ].l J

j 1, .•. , 2n+ 1 •

For m > 1 , and arbitrary r (1 2 r 2 m), T(x) is

the unique trigonometrie polynomial of order n whieh

minimizes

(5.4) m-1

max w. I L a (j ) 0 . I j=1, ..• ,2n+2 J 0 ].lr l1+J

over all trigonometrie polynomials of order n, where

w. are given by (3.13), J

Averaging Interpolation

6. A SPECIAL POINT SET X.

In this section we restrict attention to sets of

points X (not necessarily real) which satisfy a

linear recurrence relation X. + y ~

(i = l, ... ,n-l) . In order that the points X =

207

{xl, .•• ,Xn } be distinct, we must impose the conditions:

(ß-l)xl + Y f. ° (6.0) and

if ß f. l, then ßj = 1 for j = 2, ... ,n-l

In return for this relatively special choice of the

sequence X, we can now get results analogous to those

of Theorem 1 with only minor restrictions on the zeros

of the polynomial A(z) of Theorem 1 instead of the

requirement that the zero be negative. At the same

time, we can combine information on the vectors

+ +(k) A(E) f, ..• ,A(E) f· to get a more general interpolation

result.

THEOREM 4. Let X= {xl'···xn } with xi +l = ßXi + Y

m (i = l, ... ,n-l) satisfy (6.0). Let A(z) = TI (l+a z)

1 r

be a po ZynormaZ wi th A(ß j ) f. 0., j O,l, ..• ,n-m-l.

208 T.S. MOtzkin & A. Sharma & E.G. Straus

Then thepe is a unique poZynomiaZ P(x) of degpee

< n-m-1 such that

(6.1) A(E)P = A(E)f

PROOF: It suffices to consider the case where

A(E)f = o. Then A(E)P = 0 leads to the successive

equations:

(6.2) (J+a E) m

=

-+ (J+a1E)P = (J+amE)

= (J+a E)Q = Q = 0 m m-1 m

where Qv(x) = Qv_1(x) + avQv_1(ßx+y), v = 1, ••• ,m,

Q (x) = P(x). Since Q (~) is a po1ynomia1 of degree o m -+

< n-m-1 and since 0 = (0 (x1), ••• ,Q (x » = 0 it 1n 1n m n-m

fo11ows that Qm(x) = o. If Qm_1(x) ~ 0 and if

then from (6.2), we have

Averaging Interpolation 209

so that a ßk + 1 = 0 , i.e., A(ßk) = 0 , contrary to m

hypothesis. Hence Q l(x) = 0 • m-Simi1ar reasoning

gives Qm_2(x) = ... = Q1(x) = P(x) = o. This

comp1etes the proof of the theorem.

THEOREM 5. x = ßXi + y i+1

(i = 1, ••. ,n-1) satisfying (6.0). suppose

A(ßj ) I 0, j = O,l, ••• ,k(n-m)-l

where k is a given integer ~ 1. Then there exists

a unique poZynomiaZ P(x) of degree ~ k(n-m)-l such

that

A(E)P(j) = A(E)1(j) , j = 0,1, .•. ,k-1

where

= (f (j ) f (j ) ) 1 ' ... , n

210 T.S. Motzkin & A. Sharma & E.G. Straus

PROOF: As in the proof of Theorem 4, it suffices to

consider the case f = 0 which leads to Q (j ) (x.) = 0, m 1

i = 1, ••• ,n-m; j = 0,1, ••• ,k-1 Since ~(x) is a

po1ynomia1 of degree < k(n-m) with k-fo1d zeros

at it follows that Q (x) = 0 . m

By

the same argument as in the proof of Theorem 4, this

imp1ies that

Qm_1(x) - - Q1(x) = P(x) - 0 •

Remark 1: If the condition A(ß j ) # 0 is violated

for some j € {0,1, ••• ,n-m-1} in Theorem 4 or for

j E {O,l, ••• ,k(n-m)-l} in Theorem 5, say a = 1 -j

-ß

then the condition Q1 - 0 does not imp1y that

p(x) = 0

arbitrary

If A(z)

then

Indeed P(x) = c{(ß-1)x + y}j with

c satisfies Q1(x) = P(x) +a1P(ßX+Y~ = 0 • j1 Jk

has severa1 distinct zeros ß , .•• ,ß ,

P(x) k . J v = lC {(ß-1)x+y} 1 v

Multiple zeros of A(z) of the form ßj do not lead

to additional free parameters.

Averaging Interpolation 211

Remark 2: We can ca1cu1ate the po1ynomia1 P(x) of

Theorem 4 exp1icit1y by a simple device. Let

l(x) =

be the po1ynomia1 determined by the conditions

Then

(6.3) A(E)P(x) = l(x) .

If ß # 1 ,put Y = x + ~ and set Q(y) =

P(y - ß~l) , where

n-m-1 Q(y) = L

o

Then (6.3) yie1ds

that is,

212 T.S. Motzkin & A. Sharma & E.G. Straus

n-m-1 c A(l) + \ c A(ßv)yV = t(y - -1-)

o L v ß-1 o

=

n-m-1 t (v) (-1-) \ 1-8 v Lv! Y o

Hence

c = t(V)(y/(l-ß» v v!A(ßV)

v = O,1, •.• ,n-m-1 •

If ß = 1 , the problem of finding P (x) becomes

simpler. Indeed, putting E = 1 + ß , where

ßf = f(x+1) - fex) and ßV = ßßv-1 , and setting

A(l+x)

we have

P(x) 1

= A(l+ß) tex) .

If

00

-1 \ v (A(l+x» = Lbvx , o

we have

Averaging Interpolation 213

p(x} =

7. RELATIVE UNISOLVENCE.

In this section we discuss the concept of relative

unisolvence introduced in [3] and extend it to finite

dimensional subspaces which are not necessarily of

co-dimension one.

DEFINITION. Let F be a Zinear spaae and Zet

L = {L.li € I} where I is some index set, be a 1

maximaZ system of ZinearZy independent funationaZs on

F so that f € F is determined by its ao-ordinates

i € I . An n-dimensionaZ subspaae F of F n

is unisoZvent reZative to L if an eZement f € F

is determined by any n of its ao-ordinates

Li f, ... ,Li f where {il, ... ,in} cl. 1 n

The examples given in [3] illustrate this general

definition when the finite dimensional subspace is of

co-dimension one.

214 T.S. Motzkin & A. Sharma & E.G. Straus

Example: Let F be the space of functions from a field

A to itself and let L = {L la E A} where L f = f(a) a a

for all a E A and all f E F • If 'JT n-l is the

subspace of polynomials of degree < n-l in A[x] -then by Lagrange interpolation, we know that f E 'JT n-l

is uniquely determined by L f, ••• ,L f al an

for any

Thus is unisolvent relative

to L.

On the other hand if we let F be the space of

formal power-series A[[x]] and let L = {Lo,Ll , ••• }

where L f = coefficient of n

n x in f , then the

subspace 'JT n-l

of polynomials of degree < n-1 is not

unisolvent relative to L since Lnf = Ln+1f = = 0

for all f E 'JT 1. n-

LEMMA 1. The space F is unisolvent relative to n

if and only if for any n + 1 functionals

L ,Ll, .•• ,L E L ~ there is a unique linear relation o n

(7.1) (L + alL 1 + ... + a L )f o n n o

L

Averaging Interpolation 215

for aZZ f E F ~ where the n

aroe non-zero saaZaros.

PROOF. It is c1ear that (7.1) imp1ies uniso1vence

relative to L. For, let L1 , ••• ,Ln be any n-tup1e

of functiona1s of L, then (7.1) shows that L1f, ••• ,

L f n

determine L f o

for any L E L o

and hence

determine f.

Converse1y, assume F unisolvent relative to n

Since F is n-dimensiona1, any n + 1 functiona1s n

L •

L , ••• ,L restricted to F satisfy a linear relation o n n

b L + ... + b L = 0 Now if one of the b. were o 0 n n l.

zero (say b = 0) o '

then there wou1d be a linear

dependence among L1 , .•. ,Ln on Fand if, say, n

wou1d determine L fand n

hence f for all f E F contrary to the fact that n

Fn is n-dimensiona1. Hence all the bi are non-zero.

Since b ~ 0 , we may assume b = 1. If there o 0

were two different relations of the form (7.1), then

e1iminating L o among them wou1d lead to a linear

dependence among L1 , ..• ,Ln again contradicting the fact

216 T.S. Motzkin & A. Sharma & E.G. Straus

that F is n-dimensiona1. This comp1etes the proof n

of the lemma.

THEOREM 6. The space nk_1 of poZynomiaZs of degree

< k < n-m is unisoZvent reZative to the functionaZs

L1 , •.• ,L of Theorem 2. n-m

PROOF. For the sake of simp1icity we prove the

theorem for the functiona1s of Theorem 1. We need to

show that for any k functiona1s

conditions

L. , •.. ,Li the 1.1 k

(7.2) = L. P = 0 1.k

app1ied to the vector with

P = P(x), v = 1, .•. ,n imp1y p(x) = 0 • \! V

As in the proof of Theorem 1 we write

A(E) = (J+u1E)A1 (E) and use the fact that the

-+ app1ication of A1 (E) to P is variation diminishing.

-+ -+ Set A1 (E)P = (Q1, •.• ,Qn-m+1) = Q. If Q is not

zero then the conditions (7.2) imp1y

Averaging Interpolation 217

= 0

= 0

-+ which in turn imp1ies that P has at least k sign-

changes so P(x) = 0 • -+

If Q = 0 , then we write

and proceed as before.

COROLLARY. L. , .•• ,L i are any 1.1 k

Suppose k func-

tionals of Theorem 2. Then there exist non-zero con-

stants (unique except for a constant factor) c1 , ... ,ck such that

(7.3) 0, f E: 1Tk_2 .

PROOF. From Theorem 6 the space 1Tk_2 is unisolvent

relative to L1 , ••• ,L and from Lemma 1, we know n-m

that there exist non-zero constants c satisfying v

the coro11ary.

218 T.S. Motzkin & A. Sharma & E.G. Straus

In order to find the exp1icit va1ue of c , we \)

observe that the po1ynomia1 P(x) € ~k-1 which

satisfies the interpo1atory conditions

L. (P) = Li (f) , \) = 1, ••. ,k, 1v \)

is given by the fo11owing determinantal equation:

1 2 k-1 P(x) x x x

Li (1) Li (x) 2 Li (xk- 1) L. (f) Li (x )

1 1 1 1 ~1

Li (1) L. (x) 2 Li (xk- 1) Li (f) L~ (x )

k ~ k k

Then

(7.4)

1 k-1 x x

= 0

0

(_1)k+1M•P(x) = Li (1) L. (x) Li (xk- 1) Li (f) 1 ~1 1 1

Li (1) Li (x) L. (xk- 1) L. (f) k k ~k ~k

Averaging Interpolation

where

M ,. ~ 0 by Theorem 6.

If f is a polynomial of degree ~ k-2 , it is

clear that P(x) = 0 , hence the coefficient of

219

k-l x in the determinant on the right in (7.4) vanishes

when f E nk_2 • Thus

Li (1) 1

Li (x) 1

Li (xk- 2) 1

Li (f) 1

(7.5) = 0

L~ (1) L~ (x) Li (xk- 2) k

L~ (f)

Since the determinant M ~ 0 the minors of its last

column are not all zero, so that (7.5) is a non-trivial

linear r ela tion among the Li (f) , i = 1, ••• , k •

Therefore by the corollary all the minors are non-zero

220 T.S. Motzkin & A. Sharma & E.G. Straus

and so the relation (7.5) coincides with (7.3). Hence

L, (1) 1 1

L, (x) ~l

L (1) L. (x) L, (xk- 2) i, 1 1, 1 1, 1 J- J- J-

e, = J L. (1) L (x) L (xk- 2)

1 j +l i j +l i j +l

L, (x) 1 k

k L, (x )

1 k

On the basis of Lemma 1, we see that if F is of

dimension (N > n) that is; F = {(fl , ... ,fN)} , and

if L = {LI"" ,LN} with

for all f ~ F , then F n

L,f = f., (i = 1, .•• ,N) 1 1

is unisolvent relative to

L if and only if it intersects each of the co-ordinate

(N-n)-subspaces only at the origin.

The best approximation problem settled in (I3],

§§2 and 3) can now be extended to the more general

situation discussed here.

Averaging Interpolation

PROBLEM: Let A be a valued field and let F be a

spaae of dimension N with aoordinates given by the

maximal system of linearly independent funationals

L = {Ll, ... ,LN}. Suppose Fn is a subspaae of

dimension n « N) unisolvent relative to L. Por

any cP E: F " find the

distanae funation

f E: F whiah minimizes a n

(7.6) Ilcp - fll = H(IL1(CP-f)I , ... ,I~(CP-f)l) •

221

where H is some (aonvex) funation of N non-negative

variables.

To illustrate the method we restriet attention to

the case when A = R , the real field and

the Euclidean distance.

By unisolvence there exist unique elements

f(l) , ... ,f(n+l) E: F so that n

L f(i) = L j j

j f i, j = l, ... ,n .

222 T.S. Motzkin & A. Sharma & E.G. Straus

If ~ e F , our problem is trivia11y solved by f = ~. n

If ~ t F , then by uniso1vence every f e F has a n n

unique expression

(7.7)

Hence

n+1 f = L A f(i)

1 i

n+1 L A. = 1 • 1 ~

We can now extremize (7.3) subject to the condition

to get

(7.9)

n+1 L A = 1 1 i

(i = 1, ••• ,n+1)

Averaging Interpolation 223

where ~ is the Lagrange multiplier determined so that

and

(7.10) ~ij =

n+l I Ai 1, 1

The equations (7.9) determine the Ai and hence f

uniquely.

8. APPLICATION.

Theorem 1 shows that if A(z) of (2.1) is a

polynomial having only negative zeros, then polynomials

of degree ~ n-m-2 are unisolvent relative to the n - m

functionals

L (f) \)

m - 1a fex + )

o ~ ~ \) \) = 1, ... ,n-m •

It follows from Lemma 1 (§7) that there exist non-zero

scalars ll, ... ,l such that n-m

224

(8.1)

T.S. Motzkin & A. Sharma & E.G. Straus

n-m ( I l L )P = 0 ,

1 v v P E 'IT 2. n-m-

We can now app1y Lemma 1 in [3] (p. 1200) and obtain

the fo11owing theorems:

THEOREM 7. Fop any funation CP.t the poZynomiaZ P(x)

of degpee < n-m-2 whiah minimizes

n-m (8.2) I ~kILk(cp-p)la, ~k > 0, a > 1

1

is unique and aan be wr'itten as

(8.3) n-m

P(x) = I AkPk(x) 1

whepe Pk(x) aY'e the poZynomiaZs dete~ined by the

interpoZatopy aonditions

(8.4) L. (Pk ) = L. (cp) , j '" k, j = 1, ••• ,n-m J J

and

Averaging Interpolation 225

where the lk are given by (8.1).

THEOREM 8. For any funation <I> ~ the poZynomiaZ p(x)

of degree n - m - 2 whiah minimizes

(8.5) max ~kILk(<I>-p)1 , ~k > 0, k = 1, ••• ,n-m • k

can be wri tten as

(8.6) n-m

P(x) = I AkPk(x) 1

where Pk(x) are the poZynomiaZs given by (8.4) and

(8.7)

9. WEAI< AND STRONG A-SIGN CHANGES.

If A(z) is a po1ynomia1 of degree m given by

(2.1) having all negative zeros, then we sha11 say that

a function fex) has n - m - 1 weak A-sign changes

on X if there exist n points < x n

in X

226 T.S. Motzkin & A. Sharma & E.G. Straus

such that for € = 1 or -1,

(9.1) m

€(-l)j La f{x +j) > 0 o II II

j = 1,2, ••• ,n-m •

If (9.1) ho1ds with > instead of ~, f{x) will be

said to have n - m - 1 strong A-sign changes. For

m = 0 , the A-sign changes coincide with the usua1

sign changes. F 1 d (m) 1 or m > an a = , et II II

f{x ) = (_1)r-1, 1 h r = , ••• ,n; t en r

m

La f{x +j) o II II

Thus the sequences

j = 1, ..• ,n-m •

{(_1)r-1}n· has n - m - 1 weak 1

A-sign changes for any m > 1 , where m A{z) = (l+z) ,

but for m = 0 it has n - 1 strong sign changes.

We sha11 say that P{x) weak1y (or strong1y)

AX-interpo1ates f{x) if there exist points

x1 , ••• ,xn € X such that P{x) - f{x) has n - m - 1

weak (or strong) A-sign changes on X.

Let n-m-1 = b 1 kX + ... , n-m- , k = 1, ••• ,n-m

Averaging Interpolation 227

be the po1ynomia1 of degree n - m - 1 such that

(9.2) k

= (-1) j = k

as j runs through 1, .•• ,n-m. The polynomials Bk(x)

are unique1y determined as is seen from Theorem 1. We

now formu1ate

LEMMA 2. AZZ poZynomiaZs T(x) of degree n - m - 1

whiah have n - m - 1 weak A-sign ahanges on X are

given by

(9.3)

whex>e

(9.4) n-m l Ak = 1 • 1

Fox> T(x) to have n - m - 1 stx>ong A-sign ahanges

on X ~ (9.4) is to be x>epZaaed by

(9.5) Akb 1 k > 0 , n-m- ,

n-m l Ak = 1 • 1

228 T.S. Motzkin & A. Sharma & E.G. Straus

The poZynomiaZs Bk (X)/bn_m_1 ,k are the onZy ones

which AX-interpoZate zero exactZy n - m - 1 times.

LEMMA 3. If a poZynomiaZ P(x) of degree n - m - 2

n-m-1 weakZy AX-interpoZates -x " then there exists

a set of positive numbers w1 ""'w such that n-m

P(x) minimizes the A-norm (p = 1)

(9.6) n-m m " I w. I a P (x +.)

j=l J ~=o ~ J

where P(x) = xn- m- 1 + p(x) •

PROOF. By Lemma 2, since P(x) weak1y A-interpo1ates

zero on X, we have

* where the Ak satisfy (9.4). Set

!bn- m- 1 ,j! * wj = if A. ;: 0 J

2!bn_m_1 ,j! * = if A = 0 j

Averaging Interpolation

Since

* -1 j =Ajb 1.(-1) n-m- ,J

we have

n-m m L wjl L aP(x+.)1

j=l ~=o ~ ~ J

For any other po1ynomia1

we have

n-m m LW. I L a T (x +.) I =

j=l J p=o P ~ J

=

where }:' extends over those

* f Aj :/: 0 and extends over

This proves the lemma.

n-m * = L A. = 1 •

j=l J

with

n-m

n-m L A = 1 , 1 k

IAjl L w

j=l j Ibn- m- 1 ,jl

, " L A. + 2 L Aj > 1

J

indices j for which

* those for which A = j

229

o .

230 T.S. Motzkin & A. Sharma & E.G. Straus

LEMMA 4. Let p > 1 be given. If a poZynomiaZ P(x)

of degree n - m - 2 strongZy A-inter-poZates

n-m-1 -x on X then there exist positive weights

w1 , ... ,wn_m such that P{x) minimizes the A-noPm:

where A n-m-1 P{x) = x + P{x) •

PROOF. By Lenuna 2,

A n-m * -1 * P{x) = L Akb _ -1 kBk{x) , Akb 1 k > 0 , k=l n m , n-m- ,

n-m * L Ak = 1. 1

Set

_ I IP * 1-p w. - b 1. (A.) • J n-m-,J J

Then 11 P IIA = 1. For any other po1ynomia1 T (x) , ,p

as in (9.3), we have

n-m L A. = 1 1 J

Averaging Interpolation 231

* which will have its minimum if A. =A ,which comp1etes J j

proof of the lemma.

These lemmas lead to the fo110wing theorems. (We

omit the proofs as they run parallel to those of

Theorem 13.10 and 13.11 in I51, p. 286-288.)

THEOREM 9. If P(x) is a poZynomiaZ of degree

n - m - 1 whioh weakZy A-interpoZates f(x) on X

(having M points), then there is a set w. J

(j = 1, ••• ,M-m) of positive numbers suoh that P(x)

is a best approximation to f(x) in the weighted

A-nomz (p = 1)

M-m m L w.1 L a {p (x +j) - f (x]1+j)} 1 •

j=l J ]1=0 ]1 ]1

THEOREM 10. If in Theorem 9, the "weakZy" is repZaoed

by "strongZy", then for any p > 1 , there is a set

w > 0 j

(j = 1, ... , M-m) suoh that P(x) is a

best approximation to f(x) in the weighted A-nomz:

lM-m m J1/P L wj l I a {P(x +.) - f(x +j)}I P , j=l ]1=0]1 ]1 J ]1

p > 1 .

232 T.S. Motzkin & A. Sharma & E.G. Straus

10. CONCLUSION.

the

It may be interesting to remark that if X is

th n row of a triangular matrix with all the

in [a,b] , then for any given m and fixed polynomials

Ai(z) a Faber-Bernstein type result holds. In fact

using the argument in [1], it is easy to see that if

IDfx (xi +l - Xi) tends to zero as n + ~ , there exists

a continuous function f such that the polynomials

P(x) of Theorem 2 do not converge to f(x) • It

would be interesting to see if other results for

Lagrange interpolation hold for the polynomials of

averaging interpolation as weIl.

1. Curtis, P.C.: polynomials. pp. 385-387.

REFERENCES

Convergence of approximating Proc. Amer. Math. Soc. 13 (1962),

2. Malozemov, V.N.: On the method of equal sums (Russian) Vestnik Leningrad Univ. 13 (1967), pp. 167-170.

Averaging Interpolation 233

3. Motzkin, T.S. and Sharma, A.: Next-to-interpo1atory approximation on sets with mu1tip1icities. Can. J. Math. 18 (1966), pp. 1196-1211.

4. Motzkin, T.S. and Wa1sh, J.L.: Least pth power polynomials on a real finite point set. Trans. Am. Math. ·Soc. 78 (1) (1955), pp. 67-71; 83 (2) (1956), pp. 371-396.

5. Rice, J.R.: The approximations of functions Vo1. 11, Addision-Wes1ey (1969).

ON THE CONSTRUCTION OF MULTIDIMENSIONAL SPLINES

Marie-Jeanne Munteanu

INTRODUCTION

Smoothing polynomial splines have been in­

troduced by I.J. Schoenberg [17]. Generalizations

of this notion have been given for example by

M. Atteia [2], [3], G.M. Nielson [15].

In a previous paper we have given a general

abstract definition of splines for the combined case

of interpolation and smoothing (see [14]).

The purpose of this paper is to give several

methods of construction for some important classes of

splines which are particular cases of the general

definition mentioned above.

235

236 M. Munteanu

1. GENERAL DEFINITION.

Let X,Z be Banach spaces, Zi i = 1,m+n

Hi1bert spaces.

We want to approximate Gx, x E X where G is

continuous linear operator on X into Z, using

m + n observations h i , i = 1,m+n. Suppose

j h. = Fx, j = m+1,m+n are interpo1ating da ta and

J

the elements h i , i = 1,m represent experimental

va1ues which are approximations of the quantities

F~, i = 1,m where Fi , i = 1,m+n are continuous

linear operators on X into i = 1,m+n •

Consider U a continuous, linear and surjective

operator from X onto a Hi1bert space Y. Let us

define an operator V on X into n, where n is

the cartesian product of the spaces

12 m Vx = [F x,F x, •.. ,Fx] .

1 m Z , ••• ,Z •

Denote by ZO = Yxn the cartesian product of the

spaces Y and n and we define the operator L on

X into ZO as fo110ws

Multidimensional Splines 237

Lx = [Ux, Vx] •

We note also by hO = [O,h] € ZO , where 0 is the

origin of Y and h = [hl ,h2 , .•• ,hm] € n. As Atteia,

Anselone, Laurent Il] [2] [3] did for the case of

smoothing spline for functionals we introduce in ZO

the following norm

lI[y,Zl' ••• 'Z ]11 2 m ZO

p > 0 •

DEFINITION: We wiZZ aaZZ a generaZized smoothing

spZine tor operators, any eZement s € X whiah

minimizes the quantity:

under the aondi tions i = m+l, ••• ,m+n •

Remark:

10 • This definition combines the smoothing and

interpolating case.

20 • For the case p = ~ , we obtain the generalized

smoothing splines for operators, which we treated in

238 M. Munteanu

a previous paper [13].

3°. The mentioned authors Atteia, Anse10ne, Laurent

did not treat the eombined ease. They studied

genera1ized sp1ines for the ease when Fi are

funetiona1s.

4°. The degree of genera1ity chosen permits us to

treat severa1 interesting eases for app1ieations.

We find again some sp1ines treated a1ready by Nie1son

in [15], Atteia [2], [3] and find severa1 new types of

sp1ines.

THEOREM. (Existenae and DniquenessJ

Suppose:

whe1'e i

= 1,m} N1 = {x E XIFx = 0, i

j N2 = {x E XIFx = 0, j = m+1,m+n} .

2°. U(N2) is a aZosed bounded set.

Multidimensional Sp1ines 239

Then there exists a unique SEX such that

= Min i i

Fx=h ,i=m+ 1, m+n

For the proof of the theorem see the paper I presented

at the meeting on Approximation Theory, Michigan State

University, 22-24 March 1972. ([14]).

2. METHODS OF CONSTRUCTION.

The purpose of this paper is to present three

different methods for the construction of multidi­

mensional sp1ines.

I. Variationa1 method. In a preivous paper [11],

we genera1ized for the one dimensional case a

variationa1 method given by Carasso in his doctora1

thesis [4]. The intention is to adapt this method

to the multidimensional case.

ll. The method based on Gordon's interpolation

resu1ts [5] [6], of forming minimal and maximal

projectors. for the case of more than two variables,

I studied in my doctora1 thesis intermediate projectors

240 M. Munteanu

[12]. Ana1ogous projectors will be considered in this

case. We will present these methods for classes of

splines in severa1 variables, which are particu1ar

cases of the general definition.

111. The method of Anse10ne and Laruent. We will

app1y the projection method given by the mentioned

authors for certain c1asses of multidimensional sp1ines.

3. METHOD 1.

Let us consider X the space of rea1-va1ued

functions f, which are defined on R = I'xI",

1'=[0 1] I" = [0,1] such that the fo11owing , x' y

derivatives exist almost everywhere respective1y on

R, I', Irr and have the fol1owing properties

D(i,j)f(x,y) € C(R), i < p, j < q

(p-1,j) Df(x,O) is abs. cont. ,

(p ,j) Df(x,O) € L2(I') j = 0,q-1

(i,q-1) Df(O,y) is abs. cont. ,

(i,q) 2 Df(O,y) € L (I") i =O,p-l

(p-1,q-l) Df(x,y) is abs. cont. ,

(p ,q) Df(x,y) € L2[R] •

Multidimensional Sp1ines

For all f be10nging to this space we have the

representation

f(x,y) p-1 q-1 i j (i,j)

= L L ~, ~ f(O,O) i=o j=o ~. J.

q-1 j 1 p-1 (p,j) + L ~ f ~x-~~r Df(t,O)dt

j=o J. 0 p-

p-1 i 1 q-1 (i,q) + \ x f (y-u)+ Df(O,u)du

L iT (q-1)! i=o 0

p-1 1 1 (x-t)+

+ f f (p-1)! o 0

q-1 (p.q) (y-u)+ Df(t,u)dtdu (q-1) !

241

One can prove that this space is comp1ete with respect

to the norm furnished by the inner product:

p-1 q-1 (i,j) (i,j) (f,g)x = L L f(O,O)g(O,O)

i=o j=o

p-1 1 (i,q) (i,q) + L f Df(O,u)Dg(O,u)du

i=o 0

242

q-1 1 (p,j) (p,j) + L f Df(t,O)Dg(t,O)dt

j=o 0

1 1 (p,q) (p,q) + f f Df(t,u)Dg(t,u)dtdu

o 0

M. Munteanu

p-1 q-1 (i,j) (i,j) = L I f(O,O)g(O,O) + [f,g]x •

i=o j=o

We noted by [f,f]x a seminorm with the null space

{x~yj}p-1 q-1. For the proof of these assertions i=o j=o

see A. Sard [16] and L. Mansfie1d [10]. Indeed X

admits the fo11owing decomposition

Let U be the operator

Uf = {(p,O) (p,l) (p,q-1) (p,q) Df(t,O),Df(t,O) .•. Df(t,O) ; Df(t,u);

(O,q) (l,q) (p-1,q)} Df(O,u),Df(O,u), ••• ,Df(O,u)

and Y the Hi1bert space

Multidimensional Sp1ines

The kerne1 of the operator U is the set

{xi i}p-1 q-1 • Y i=o j=o

Remark: I have chosen the Tay10rian functiona1s in

order to simp1ify the presentation.

The Fi are functiona1s linear and continuous

on X of the form

1 1 L(f) = I J J

i<p 0 0 j<q

(i ,j) ij f(t,u)dll (t,u)

243

+ I 1 (i,j) "j

J f(t,ß)d1l 1 (t) + I 1 (i,j) i"

J f(a,u)dll J(u) •

where each

i<p 0

i,j II

j<q 0

is of bounded variation on its

domain. It is a natural way to deduct from the

variationa1 definition, using ca1cu1us of variations,

the ana1ytica1 expression of the sp1ine function s.

App1ying the method which I treated in one dimensional

case in [11], we obtain in a simi1ar way

244 M. Munteanu

s(x,y) = p-1 q-1 . . i. m+n . l l a~,Jx yJ + l A.F~ [K(x,y,t,u)]

i=o j=o i=l ~ u

where the coefficients fo11ow from the conditions

i = 1,m

j = m+1,m+n

where

K(x,y,t,u) = (G(x,y,~,n),G(t,u,~,n»y •

G(x,y,t,u) is defined as:

p-1 q-1 i j (i,j) f(x,y) = l l ~, ~ f(O,O) + (G(x,y,t,u),Uf(t,u»y.

i=o j=o . J

The function K(x,y,t,u) is the reproducing kerne1

of the orthogonal comp1ement of the set {xi yi}p-1,q-1 i=o j=o

which is the null space of the operator U.

Multidimensional Splines

q-l p-l i t i 1 (y-n)+

K(x,y,t,u) = L ~! i! f (q-l)! i=o 0

q-l (u-n)+ (q-l) ! dn

p-l q-l j j 1 (x-S)+

+ L ~ ~, f (p-l)!

p-l (t-S;)+ dS; (p-l) ! j=o J. J. 0

245

1 1 p-l p-l q-l q-l + f f (x-O+ (t-S;)+ (y-n)+ (u-n)+ cl;dn

(p-l) ! (p-l)! (q-l) (q-l) I o 0

One can verify that the resulting spline function s

minimizes p(f) under the restrietions F~ = Fl = Ljf

= hj j = m+l,m+n. Indeed one finds that

p(f) - pes) ~ 0 , where

o 0 i=o 0

See for proof. G. Nielson [15].

The uniqueness is assured by the hypo thesis

246 M. Munteanu

Ker U N2) = 0 X ' and we suppose also that

i i i {F [K(x,y,t,u)], i = 1,m+n; x Y , i = 0,p-1, j tu

is a 1inear1y independent set.

4. METHOn 2.

0,q-1}

A very important case will be when the operators

Fij = L iM j are product of linear continuous func­x y

tiona1s L i defined on the one dimensional space x

defined on the space

We note by S the genera1ized sp1ine in one x

variable relative to the operator nP and the

functiona1s Li. x

We set x = HP[O 1] , x' 2 Y = L [0,1] , x

U = nP , and we suppose that the hypothesis which assures

the existence and unicity of S is valid (see the x

general theorem of the section 1).

Similarly we introduce the operator S Y

The

Multidimensional 8plines

operators 8 and 8 are projections. x y Using

Gordon's procedure (see {5] [6]) we can form the

analogue of minimal approximation for the case of

two variables.

81 = 8 8 • x Y

We consider the same spaces as in §3. The spline

function 81 will minimize the quantity ~(f) under

247

the conditions Fjs = hj j = m+l,m+n. (~(f) defined

as in section 3).

Let us form now the analogue of maximal

approximation:

We choose the operator U being n(p,q) , the space

being

R = [0,1] x [0,1] , the operators F x y

(a,O) f(x,y) + f(xi,y)

(O,ß) f(x,y) + f(x,y.)

J

a < p

ß < q

248

(a ,ß) f(x,y) ~ f(xi'Yj)

Corresponding to the operators

M. Munteanu

a < p, ß < q •

ia F we choose the x

spaces Zi being the space of functions fex) defined

on [0,1] such that the p-th derivative exists almost x

everywhere on [0,1] ,is an element of x

and such that the Tay10r formu1a

fex)

p-1 p-1 i (i) 1 (x-t)+ (p)

= L ~, f(O) + f (p-1)! f(t)dt, i=o . 0

2 L [0,1] , x

X E [0,1] , x

is valid. This is the Sobo1ev space HP [O,l] • x

Corresponding to the operators ·ß

FJ we choose the space y

zj being the Sobo1ev spaces Hq [O,l] • Obvious1y y

corresponding to the functiona1s FiaFjS Zij will x Y ,

be the euc1idian space R.

The quantity which is minimized by S2 is the

following

Multidimensional 8plines

1 1 1(f) = f f (D(P,q)f)2dtdu + l l (FiaFjßf

o 0 ia jß

1 + l f [Fia(Dqf) - hia]2du

ia 0

+ l fl[~ß(DPf) - h jß ]2dt • jß 0

One can easily prove that ~(f) - ~(82) ~ 0 • «see

G. Nielson [15]). For the three dimensional case we

can form for example

888 x y z

corresponding to the minimal approximation,

82 = 8 + 8 + 8 - 8 8 - 8 8 - 8 8 + 8 8 8 x y z x y x z y z x y z

corresponding to themaximal approximation,

83 = 8 8 + 8 8 + 8 8 - 28 8 8 x Y x z Y z x Y z

corresponding to an intermediate approximation.

In my doctoral thesis [12] I treated the inter­

mediate approximations including the limiting cases

of minimal and maximal approximations, for the n

249

250 M. Munteanu

dimensional case. Analogously we can form the

corresponding spline functions choosing in an

i appropriate way the spaces Y, Z , the quantity

~(f) , as we did for the two dimensional case.

5. PROJECTION METHOD.

Anselone and Laurent presented the projection

method for the case of interpolation [1], and for the

case of smoothing [8]. The purpose of this section is

to adapt the projection method for the case of

smoothing splines in several variables.

We will present first the general definition

of smoothing splines given by Atteia in [2] [3] and

describe the projection method of construction as

given by Laurent in [8].

Let X and Y be two Hilbert spaces and T a

linear continuous operator on X onto Y. Denote by

N the null space of the operator T; we suppose

dirn Ker T = n

Let Li' 1 < i 2 m , where m ~ n , be con­

tinuous linearly independent functionals defined on

Multidimensional Sp1ines

X. We denote by K the subspace spanned by the

Riesz representers of these functiona1s.

We designate by z = Y x E the cartesian m

product space, endowed with the sca1ar product

251

(p > 0) •

We define the operator L on X into Z

Lf = ITf,Af]

We put a = [O,e] E Z, 0 being the null element of

Y •

We sha11 refer to a genepaZized smoothing spZine

as an element s of X which minimizes the quantity

L(s) = IITs 11; + pllAs - eil; = IILs - all~ • m

We denote by N~ and K~ the orthogonal comp1ements

of N and K.

Under the hypothesis ~ N n K = 0 , one can

demonstrate the existence and unicity of s (see

252 M. Munteanu

Atteia [2], [3]).

Now we give Laurent's resu1ts concerning the con­

struction of the genera1ized smoothing sp1ine [8].

LEMMA 1. The suhspaae H = K n N.L is of dimension

m - n •

LEMMA 2. The kernel * G of the adJoint operator L

is a suhspaae of Z of dimension m - n •

LEMMA 3. If h i -, 1 2 i 2 m - n -' form a basis of H-,

we can introduce in G the following basis

1 < i < m - n

where

1 < i < m - n •

THEOREM. Under the hypo thesis N n K.L = 0 -' there

exists a unique spline s whiah minimizes L(s) ; viz.

m-n Ls a + I Aig.

i=l l.

where the coefficients A. are the solutions of the l.

Multidimensional Splines 253

linear system

The steps of the construction are the following:

1. One seeks a base hi , l<i<m- n , of

H = K n rI m

hi = I bikj l<i<m- n • j=l

2. One deduces the base ~i

1 < i < m - n •

3. Then follows the base

4. Then one can write

254 M. Munteanu

m-n Ls - a = ): \gi

l.=1

the coefficients Ai are given by the system (*)

5. From Ls = [Ts,As] OIle has fina11y

m-n Ts = L Ai~·

i=l l.

As - e

We will app1y this method to the fo110wing c1ass of

smoothing sp1ines in two variables. Let us consider

X the space of rea1-va1ued functions, defined on

R = I'xI", I' = [0 1] , x' I" = [0 1] such that , y'

the fo110wing derivatives exist almost everywhere

respective1y on R, I' , I" , and

(x,y) ER,

D(2m-j ,j)f(x,O) E L2(I') , x E I', j < m ,

Y EI", i < m ,

Multidimensional Splines

and such that the Taylor formula

f(x,y) \' xi l. n(i,j)f(O 0) L .,. , '

i+j<2m 1.. J.

1· 1 ( ) 2m-i-I y-u + (i 2m-i)

+ .L ~! 1 (2m-i-I)! n' f(O,u)du 1<m 0

. 1 ( )2m- j -I ~ x-t + (2m-j j)

+ L ., 1 (2m-j-I)! n 'f(t,O)dt j<m J. 0

m-I m-I (x-t)+ (y-u)+ (m m)

+ 1I (m-I)! (m-I)! n ' f(t,u)dtdu R

holds for any (x,y) ER.

With the inner product

(f,g)X = I n(i,j)g(o,o)n(i,j)g(O,O) i+j<2m

1 + I 1 n(i,2m-i)f(0,u)n(i,2m-i)g(0,u)du

i<m 0

+ I IIn(2m-j ,j)f(t,0)n(2m-j ,j)g(t,0)dt j<m 0

+ Iln(m,m)f(t,u)n(m,m)g(t,u)dtdu R

255

256 M. Munteanu

X is a Hilbert space.

Thespace X admits in fact the following

decomposition

where ~ designates the number of functionals

D(i,j)f(O,O), i + j < 2m •

Let T be the operator defined by

Tf = {D(2m,O)f(t,O);D(2m-l,1)f(t,O); ••• ;

D(m+l,m-l)f(t,O);D(m,m)f(t,U);D(O,2m)f(O,u);

D(l,2m-l)f(O,u); ••• ;D(m-l,m+l)f(O,u)}

and Y the Hilbert space

We see easily that the kernel of T is the set

of polynomials of degree less than or equal to 2m - 1

Multidimensional Splines 257

in x and y Then Ker T and

dim Ker T = ~ < 00

We consider n(n ~W continuous, linearly independent

functionals of the same form as in the Section 3. We

will suppose that the subspace spanned by the Riesz

representers of these functionals contains the kernel

of the operators T. Denote by e = [el ,e2,··· ,en] € En

the vector symbolizing the experimental values of the

given functionals for f ( X •

Let us denote with s(x,y) the corresponding

smoothing spline.

In order to employ the projection method

previously explained, we must choose n - ~ func­

tionals 0., 1 ~ i ~ n - ~ , linearly independent, 1

defined on X, and having the following properties

where

the functionals

~ . I Aj Nj

1 (f) j=l

p = l,~,

to the set of given functionals,

q = l,n-~, belong

258 M. Munteanu

20 the n - ~ chosen functionals exhaust the n

given functionals.

One verifies immediately that if we choose

o (f) = q

q q q q q i j q 2m-I) q" N (I)N (x)N (xy)N (y) ••• N (x Y ) "+" 2 ••• N (y N.l.f) ~ ~ p p P 1 J< m ~ p

we have

o (f) = 0, Vf E Ker T, q q I,n-p •

Choosing these functionals we used Jerome and

Schumaker's idea for the case of univariate splines.

(see [7]).

If we appIy the functionals o , q

to the representation of f we obtain

q I,n-p,

Multidimensional Splines

o (f) q

[ , 2m-i-I~ 1 x1 (y-u)+ (i 2m-i)

L f 0 7f (2 -'-1)1 D' f(O,u)du i<m 0 q 1, m 1 •

~ , 2m-j-l~ 1 J (x-t) + (2' ') + L f 0 ~ (2 -'-1)1 D m-J,J f(t,O)dt , qJ. mJ . J<m 0

259

(y-u)+ (m m) m-I] (rn-I)! D ' f(t,u)dtdu.

We deduce the base q = I,n-)l

(Y_U)m-I~ ~(X_t)m-Il (Y_U)m-I~ -,---..,+,..-- 0 + +

(m-I) ! ' q (m-I) ! (m-I)! '

r m-I (x-t) m~ } 8 Y + q t(m-I) ! m!

E Y , q = I,n-)l

and the scaIar products

260 M. Munteanu

(x-t)+ (y-u)+ ~ m-l m-~ + ffR öq (m-l) I (m-l) I

u

Concerning the base

(x-t)+ (y-u)+ ~ m-l m-l~ öl (m-l)! (m-l)! dtdu.

b , q

q = l,n-~ , we can

obtain the components of the vectors by expanding the

functionals Ö, q = l,n-~ according to the elements q

of the last column.

Having the bases ~l' l = l,n-~ and

b l , l = l,n-~ we can write the base gl' l = l,n-~

Then we have

n-~

Ls - a = lAg q=l q q

Mu1tidimensiona1 Sp1ines 261

where Lf = [Tf,Af] € Z, Z= Y xE, Af € E being

the vector having the n components {L kM . .eJk€J' ,l€J"o x J

The coefficients Aare the so1utions of the

system

a = [O,e] € Z ,

We deduce

and

q

n-J.\ Ts = L A <P

q=1 q q

l = 1,n-J.\

262 M. Munteanu

Fina11y in order to find s(x,y) we use the

representation (1)

s(x,y) =

Remarks:

i j (. .) L .;-~ D 1.,J s(O,O) .+. 2 1. J. 1. J< m

i 1 + L ~J

i<m 1.. °

. 1 ~ + L . ,1

j<m J. °

T)-].l + L JJ

q=l R u

(y-u) 2m-i-\_].l ~ i ( lm-i~l + \' A 0 ~ -..,.y_-_u'-":+---'-l-

(2m-i-1)! q:1 q q i! (2m-i-1)! du

2m-j-1 ~ (x-t) T)-].l j + 0 L

(2m-j-1)! I q j! q-1 -

m-1 m-1 (x-t)+ (y-u)+

(m-1) ! (m-1) !

(x-t) 2m-j -ll

(2m-~-1) ! j dt

~x_t)1l1.-1 o +

·q_(m-1)!

m-1~ (y-u)+ (m-1) ! dtdu.

1°. This method can be app1ied also for the case of

smoothing sp1ines described in the section 3.

2°. The construction of the genera1ized sp1ines in

severa1 variables treated by Atteia in his doctora1

Multidimensional Sp1ines

thesis [3], can be treated in an ana10gue manner.

It is c1ear that the sp1ines introduced by Atteia

263

are a particu1ar case of the general definition given

in §1.

30 . We can also app1y to the c1ass of smoothing sp1ines

treated in this section the variationa1 method of Section

3. In fact in this case we can consider the more general

case of interpolation plus smoothing. We obtain again

the ana1ytica1 expression of the sp1ine function given

by Nie1son in his doctora1 thesis [15].

REFERENCES

1. Anse10ne P.M. and Laurent, P.J.: A general method for the construction of interpo1ating or smoothing sp1ines-functions, Num. Math., 12,1968, 68-82.

2. Atteia, M.: Fonctions-sp1ine genera1isees, C.R. Acad. Sei. Paris, t. 261, 1965, 2149-2152.

3. Atteia, M.: Theorie et app1ications des fonctions­sp1ines en analyse numerique, These, Grenob1e, 1966.

4. Carasso, C.: Methodes numeriques pour l'obtention des fonctions-sp1ine, these de 3-eme cyc1e, Universite de Grenob1e, 1966.

264 M. Munteanu

5. Gordon, W.J.: Sp1ine-b1ended interpolation through curve networks, J. Math. Mech. 18, (1969), 931-952.

6. Gordon, W.J.: Distributive 1attices and the approximation of mu1tivariate functions, in Approximation with special emphasis on sp1ine functions, I.J. Schoenberg, ed. Academic Press, N.Y., 1969, 223-277.

7. Jerome, J. and Schumaker L.: A note on obtaining sp1ine functions by the abstract approach of Laurent, MRC Technica1 Summary Report #776, August 1967, Madison, Wisconsin.

8. Laurent, P.J.: Representation de donnees experimentales a l'aide de fonctions sp1ine d'ajustement et evaluation optimale de fonctionne11es 1ineaires continues, Ap1ikace Math., 13, 1968, 154-162.

9. Laurent, P.J.: Cours de theorie de l'approxima­tion, Fascicu1e 3, Facu1te des Sciences de Grenob1e, 1967-1968.

10. Mansfie1d, L.E.: On the optimal approximation of linear functiona1s in spaces of bivariate functions, SIAM J. Num. Anal. 8 (1971), 115-126.

11. Munteanu, M.J.: Observatii asupra solutiei optimale a unei probleme diferentia1e ne1iniare cu va10ri 1a 1imita, pe subspatii de functii sp1ine genera1izate, Bu11. Sei. lnst. Po1ytech. C1uj, 1 (1968), 47-56.

Multidimensional Sp1ines 265

12. Munteanu, M.J.: Contributions a 1a theorie des fonctions sp1ines a une et a p1usieurs variables, Doctora1 thesis, Univ. de Louvain, Be1gium, January 1971.

13. Munteanu, M.J.: Genera1ized smoothing sp1ine functions for operators, SIAM J. Numer. Anal. Vo1. 10, No. 1, March 1972.

14. Munteanu, M.J.: Multi-dimensional smoothing sp1ine functions, Symposium on approximation theory and its app1ications, March 22-24, 1972, Michigan State University.

15. Nie1son, G.M.: Surface approximation and data smoothing using genera1ized sp1ine functions, Doctora1 thesis, Univ. of Utah, June, 1970.

16. Sard, A.: Linear Approximation, Am. Math. Soc., Providence 1963.

17. Schoenberg, I.J.: Sp1ine functions and the problem of graduation, Proc. Net. Acad. Sei., 52, 1964, 947-950.

ON ERROR ESTIMATES APOSTERIORI IN ITERATIVE

PROCEDURES

A.M. Ostrowski

1. 1939, in a note in the C.R. of the Paris

Academy, 209, 777-779, I introduced the concepts of

the computation errors apriori and aposteriori, the

first being estimated be fore the beginning of the

essential computation and the second being deduced

after severa1 steps of the computation have been

comp1eted.

If in ametrie space the sequence x tends to v ~ the estimate aposteriori of the error, /xv'~/'

has to be deduced using the values a1ready computed

of x1 , ••• ,xv and, if possib1e, of the correction

/xv ,xv+1 / .

267

268 A.M. Ostrowski

Usua11y we have the situation where for a q,

o < q < 1 ,

(1) (v + (0) •

However, this eannot be used for the estimate a pos­

teriori of Ix ,si . v

The situation is eomp1ete1y different if we have

the recurrent estimate,

In this ease we have

(3) 1 ---<

l+q-Ixv,sl 1

Ixv,xv+1 1 ~ 1 - q

(3a)

and the estimate (3) gives obviously a pretty e10se

evaluation of Ix ,si . v

Error Estimates aPosteriori

2. We have a more general situation if (2) is

rep1aced with

(4) (\! = 0,1, ... )

where ~(x) is positive and < 1 with x > 0 and

non-increasing with x ~ o. In this case we have

(5)

269

where ~(Ix\!,sl) has to be rep1aced in each case with

a convenient majorant.

An important special case is if for a sequence

00

~\! with 1 > ~\! > 0 , TI ~ \! = 0 , we have \!=1

(6) (\! = 1,2, .•. ) .

Then it fo11ows

(7) 1 Ix\!,sl

1 + ~\! < IX\!+l,xl 1

2. -1---~- . \!

For instance we cou1d use the sequence 1

~\! = 1 - v

270 A.M. Ostrowski

3. In the case tha t x and 1;; are n-dimen­v

siona1 vectors, n > 1 , and generally xV +1 = Av Xv

with quadratic matrices A , the fo110wing estimate v

can be found in the literature *):

(8)

which can be used if the norm "Av " corresponding

to the chosen vector norm, is < 1 .

4. In the above case we have for the error

vectors ~ : = X - 1;; the relation v v

(9) t: v+1 = A ~ v v (v 0,1, •.• ) .

If A = A does not depend on v the iteration v

is convergent for any choice of ~o iff AA' the

*) Cf. for instance, J. Weissinger, Ueber das Iterationverfahren, ZAMM 31 (1951), p. 245.

Error Estimates aPosteriori

speetral radius of A, is < 1. And in this ease

we have *)

(v -+ 00) •

But this again eannot be used for estimates a

posteriori.

However, it follows from (9)

C = (I _ A )-l(~ C ) ~v v v - ~v+l '

assuming that I - A is non-singular, and therefore,

using the euelidian norms,

(10) 1 11 - A I v e

5. In order to use (10) we have to obtain

eonvenient estimates for 11 - AI, I (I - A)-ll e e

*) Cf. for instanee, A.M. Ostrowski, Ueber Normen von Matrizen, Math. Z. 63 (1955), p. 5, formula (11).

271

272 A.M. Ostrowski

for a general matrix A. To obtain such estimates

assume that the matrix A = (a ) llV

eigenva1ues AV so that AA = M~x

use the so-ca11ed Frobenius norm of

(11)

of order n has

I AI. We will v

A

n Here we have a1ways lAI; ~ L IAv I2 so that we can

v=l put

(12) t:,.A

where t:,.A i9 a "measure for the norma1ity of the

matrix A" and in particular t:,. A = 0 iff A is normal.

In this case lAie = AA •

6. In the general case we have

(13)

Error Estimates aPosteriori

(14)

The last formu1a can on1y be used if a convenient

estimate of Idet (I - A)I can be found.

7. Better estimates can be obtained if we

assume, instead, that ÄA is known and is < 1 •

273

Then we have, using convenient1y a resu1t by Henrici*),

(16)

(17)

I (I - A)-ll < 1 e-1-t. -Ä A A

*) P. Henrici, Bounds for iterates, inverses, spectra1 variation and fie1ds of va1ues of non-normal matrices, Numer. Math. 4 (1962), p. 30, theorem 3.

274

(18)

(19) I (I _ A)-ll < n ( ~A )n e - 1 - A 1 - A A A

A.M. Ostrowski

In the relations (16) - (19) AA and ~A can be

rep1aced (simultaneous1y, both in the conditions and

assertions) by arbitrary majorants as long as the

majorant of AA remains < 1 •

The simp1est majorant of ~A is of course IAI F

On the other hand, a c10se estimate of ~A in terms

of * * AA - A A due to Henrici*) is known:

(20) H3 n - n * * ~ A 2. 12 ;I A A - AA I F •

*) 1.c.p. 27, formula (1.6)

Error Estimates aPosteriori 275

8. It may be of interest to observe that the

argument of sec. 1 and 2 can be genera1ized to more

general situations. If we have, for instance, instead

of (2) the so-ca11ed weakZy ZineaP convergence *),

(21) (v = 0,1, ••• )

where N is an integer > 1 , we have, instead of (3):

(22)

*) Cf. A.M. Ostrowski, Solution of Equations and Systems of Equations, 2d. edition (1966), p. 204.

277

SPLINES AND HISTOGRAMS

I.J. Schoenberg

INTRODUCTION.

In [3] Boneva, Kendall and Stefanov (B.K.S.) have

effectively rediscovered the essential features of what

I like to call cardinal cubic spline interpolation.

Moreover, and this is an important point, the data are

not the usual function values that are to be inter­

polated, but rather approximations of the derivative

(i.e. the unknown density function) in the form of a

histogram. This (pershaps only apparent) difference is

bridged by the ingenious area-matching condition.

In [10] I carried out a suggestion of J.F.C. Kingman

Sponsored by the United States Army under Contract No. DA-3l-l24-ARO-D-462.

278 I.J. Schoenberg

(see [3, 55]) and applied variation diminishing cubic

spline approximations to histograms. Now I believe that

this approach smoothes the data too strongly and the

MRC Report #1222 is not going to appear elsewhere.

Actually, the formation of a histogram is already a

strong form of smoothing (or fluctuation-reducing) and

further smoothing should be done with care. Here we do

no further smoothing and area-matching (or volume­

matching in the bivariate case) is done exactly.

In the present paper I am describing the application

of finite spline interpolation (S.I.) to histograms in

one and in two dimensions. The results used from uni­

variate S.I. are weIl known. The results concerning

bivariate S.I. seem to be new (Theorems 3 and 4 below)

in spite of the lively activity in this field (see [1]

and [6], also for further references). This is perhaps

not surprising in view of the novel statistical setting

of these problems on the one hand, and the number of

possible variations on the theme of bivariate S.I. on

the other.

In the present paper probability considerations

and criteria are conspicuous by their absence. A

mathematical analyst can provide the statistician with

new tools of approximation. The statisticians must

decide on their usefulness and their reliability at

Splines and Histograms

different levels of probability.

I. THE UNIVARIATE CASE

1. THE MAIN RESULTS.

Let

(1.1) H = (h.), (j = 1,2, •.• ,m) J

279

be a histogram, where h. denotes the frequency in the J

interval (j-l,j), and where observations that fall on

the common boundary of adjacent class-intervals count

1. h f h as 2 1n eac 0 t ese. The following crucial defini-

tion is due to B.K.S.

DEFINITION 1. We say that the integrable funation

f(x) , defined in [O,m], enjoys the area-matahing

property for the histogram H, provided that

(1.2) / j-l

f(x)dx = h. J

(j = 1, ... , m) •

Let the symbol AM(H) denote the alass of funations

satisfying this aondition.

In [3] B.K.S. extend the definition of h. J

to all

280 I. J. Schoenberg

integers j by setting

h = 0 if j < 1 or j > m , j

and construct a cardinal quadratic spline function a(x)

that satisfies the area-matching condition (1.2) for all

integer values of j •

There is some advantage in restricting ourselves

to the class of functions AM(H) , whose elements are

defined in IO,m] only. A ready source of potentially

useful elements of AM(H) is afforded by finite S.I.

as folIows. We associate with Hits corresponding

cumulative sequence

(1. 3)

defined by

(1.4)

F = F = (F) (j = 0,1, ... ,m) , H j

1, ... , m) •

Furthermore, let

(1. 5)

denote the class of spline functions (S.F.) of degree

k , defined in IO,m] , and having the points

x = 1,2, .•. ,m-l as simple knots. This means that the

restrietion of Sex) to [j-l,j] is a polynomial of

Splines and Histograms 281

degree not exceeding k , while k-l

Sex) € C IO,m].

Observe that if k = 1 and Sl(X) is the linear

S.F. (or continuous piece-wise linear function) such

that

(1. 6) Sl(j) = Fj , (j = O, ••• ,m) ,

then

(1. 7)

is easily seen to be a step-function whose graph is

identical with the geometric representation of H by

rectangles Rj of area h j (or height h j ) and basis

[j-l,j]. It is weIl known that the interpolatory

conditions (1.6) can also be met by elements of

Sk[O,m] , provided that k is odd (= 3,5, ••• ) , and

that appropriate boundary conditions (B.C.) are pre­

scribed for Sex) • We single out the two most useful

cases, k = 3 and k = 5 , and state the known results

as lemmas (see e.g. [8, §13]).

LEMMA 1. Thepe is a unique aubia spZine S3(x) suah

that

(1. 8) S 3 (j) = F j' (j = 0, ••• , m) ,

and

282 I.J. Schoenberg

(1.9) S'(O) = F' S'(m) = F' 3 0' 3 m '

whepe F' and F' have ppeassigned vaZues. o m

LEMMA 2. Thepe is a unique quintic spZine S5(x) such

that

(1.10) S 5 (j) = F j' (j = 0, ••• , m) ,

and

(1.11) S;(O) = F~ S"(O) = F" 5 0

S'(m) = F' 5 m

S" (m) = F" 5 m

whepe F', ... ,F" aPe ppeassigned. o m

In the present paper we discuss on1y histograms H

that may be polymodal, but are assumed to be "be11-

shaped" in the sense that they have thin tai1s. Accor­

ding1y, we shaZZ assume that h1 and hm aPe smaZZ

compaped to max h.. For this reason we sha11 assume l.

in our present app1ications that all boundary va1ues

become

(1.12)

F(s) , are taken to be zero, and (1.9), (1.11), m

S3(0) = 0, Sj(m) = 0

8p1ines and Histograms 283

and

(1.13) 8;(0) = 8S(0) = 0, 8;(m) = 8S(m) = 0 .

The reason for this choice of vanishing boundary va1ues

will become c1ear in our discussion fo11owing Theorem 2

be1ow.

If the sp1ine function 8(x) satisfies (1.8), or

perhaps (1.10), it fo11ows that its derivative

(1.14) er (x) = 8' (x)

is an element of the c1ass AM(H) , for

(1.15) / er(x)dx = / 8'(x)dx 8 (j) - 8 (j -1) j-1 j-1

F. J

F. 1 = h. J- J

(j 1, ... ,m) ,

by the very definition (1.4) of the cumu1ative sequence

(Fj ). From Lemmas 1 and 2 we therefore get the

fo11owing theorems.

THEOREM 1. There is a unique

such that

284 LJ. Schoenberg

(1.17)

THEOREM 2. There is a unique

suah that

(1.19)

PROOFS: Invoking Lemmas 1 and 2 it suffices to set

(1. 20)

in \-!ew of (1.15). It is also c1ear that (1.17) and

(1.19) are imp1ied by (1.12) and (1.13), respective1y,

in view of the definitions (1.20).

Fo11owing B.K.S. we ca11 02(x) and 04(x) the

histosp1ines of degrees 2 and 4, respective1y. Observe

that the B.C. (1.17), (1.19) tend to give their graphs

the required beZZ-shaped ahaY'aater.

It might not be irrelevant to mention some optimal

properties enjoyed by the histosplines. They fo11ow

from known properties of S.I. and may be stated as

Sp1ines and Histograms

folIows.

COROLLARY 1.

1. The quadratic histospline a2(x) of Theorem 1

minimizes the integral

m (1. 21) J

o

2 (a' (x» dx

among all functions a(x) that belong to AM(H) and

satisfy the B.C. (1.1?).

2. The quartic histospline a4(x) of Theorem 2

minimizes the integral

m (1. 22) J

o

2 (a"(x» dx

among atz functions a(x) that belong to AM(H) and

satisfy the B.C. (1.19).

Of course, we a110w for competition on1y such

a(x) for which the corresponding integrals (1.21),

(1.22), make sense.

285

286 I.J. Schoenberg

2. A FIRST CONSTRUCTION OF TRE QUARTIC HISTOSPLINE

G4(x) •

Mrs. Julia Gray, of the MRC Computing Staff, wrote

a computer program, called spZint, that furnishes the

quintic spline Ss(x) and G4(x) = S;(x) of Theorem

2. The input data are the integer m (~4) and the

values of the m + 5 quantities F , •.• ,F , o m

F' , ... ,F" . o m

For statistical applications to bell-shaped distribu­

tions we choose F = 0 ,and F' = ••• = F" = 0 • o 0 m

The program uses the quintic B-spline (see [9, 70-

71]

(2.1)

where u+ = max (O,u). In terms of this function, a

general element of SS[O,m] may be written uniquely in

the form m+2

(2.2) L -2

and the program computes the coefficients c. J

for the

solution of the interpolation problem (1.10), (1.11).

Splines and Histograms

Sex) represnets an approximation of the (cumulative)

distribution function of the statistic that furnished

H. An approximation of the density function is the

histospline

m+l (2.3) °4(x) Ss(x) = L

-2

where

287

(2.4) E-oo < x < (0) •

Thus Splint solves numerically any problem (1.10),

(1.11), of so-called complete quintic S.l. for equi­

distant data. The examples given in §3 below were

computed by means of Splint. In §4 we present an

alternative numerical approach that furnishes 04(x)

direc tly, and will help to clarify our proc edure in

dealing with the bivariate case.

3. TWO EXAMPLES.

1. The age distribution of Bulgarian mothers of 1963.

From [3, 21] we take the following table giving the

age distribution of 50226 Bulgarian mothers during the

year 1963

288 1.J. Schoenberg

j 1 2 3 4

Age group

h. 7442 19261 14385 6547 J

j 5 6 7

Age group

h. 2123 451 17 J

The corresponding histogram is shown in Figure 1 where

we have changed sca1e and origin by setting

age = 15 + 5x, (0 < x ; 7) .

The frequencies h1 , .•. ,h7 are the areas (and there­

fore also the heights) of the seven rectangles of

Figure 1, of which the last does not show at our

sca1e or ordinates due to its sma11 height.

The curve shown in Figure 1 is the quartic

histosp1ine 04(x) of Theorem 2. The graph shows

c1ear1y its area-matching property. The curve owes its

nice1y ba1anced shape perhaps to the optimal property

Splines and Histograms

of minimizing the integral (1.22).

We are now going to subject our histospline to a

severe test. In [3, 23] we also find the histogram

with 7 x 5 = 35 entries corresponding to observed

*

289

* H

annuaZ frequencies. We have reproduced H graphically

also in Figure 1. How weIl does our 04(x) match the

* areas of H ? As we see from Figure 1, some of the

* areas of H are matched weIl, some less so, especially

those in the interval .8 < x < 2.2. Dur histospline

underestimates the observed high annual frequencies in

the interval .8 < x < 1.4 (ages 19-22). These obser­

vations suggest the following comments.

Dur results (Theorems 1 and 2) remain valid, up

to notational changes, if we choose unequaZ class­

intervals. Let the lengths of the class-intervals

(x. l'x.) be denoted by l. = x. - x. 1 ' while the J- J J J J-

corresponding frequencies are again h. , as before. J

The area-matching requirement now amounts to the

relations

(3.1)

x f j

f(x)dx h. J

(j = 1, ... , m) ,

290

o o o Q

c-(

1. J. Schoenberg

~~~~~--------------------~~

~--~~----------------~------------~~

o o ~

o o o -

Splines and Histograms

while the height of the rectangle R. J

is now = h.II. J J

The program Sptint is no longer applicable, and must

be replaced by the efficient programs developed by

C. de Boor (see I4] for references). These are also

based on the use of B-splines, but they use B-splines

with unequat spacing of knots.

291

We should regard our equidistant G4 (x) as purely

diagnostic. In fact Figure 1 shows that the interval

[.8, 1.4] should be chosen as one of the class-inter­

vals, in order to trap within this interval the high

frequencies peculiar to that age group. Would perhaps

a choice of class-intervals resulting in frequencies

h j that do not vary (essentially) with h , be

advantageous? More experimentation seems called for.

A last remark suggested by this example deals with

fact that the histospline G4 (x) may assume negative

values. Clearly all hj ~ 0 ; from (1.4) it follows

that (Fj ) is a nondecreasing sequence. From (1.10)

we conclude that the sequence (SS(j)) is non­

decreasing. Unfortunately, this does not imply that

SS(x) is a non-decreasing function in [O,m] In

such cases the derivative G4 (x) will assume also

292 1.J. Schoenberg

negative va1ues. In our case of the Bu1garian mothers

we find that a(1.6) = 20184.305 , whi1e at the right­

hand tai1 we have the tab1e of va1ues

x 6.0 6.2 6.4 6.6 6.8 7.0

a 4(x) 107.792 42.512 6.093 -6.513 -4.259 .000

This is due to the sma11 va1ue of h 7 = 17. Because

of the area-matching condition, this is bound to happen

in an interval (j-1,j) where hj is sufficient1y

sma11 compared to max h .• J

2. The distpibution of Zengths of eaps of cOPn. We

consider the data from I1, 93] giving the distribution

of the 1engths of 578 ears of a certain lewa variety of

corn, grouped into 14 histogram ce11s of equa1 widths,

the first ce11 representing a 1ength range of 10.5 -

11.5 cm. and the last a range of 23.5 - 24.5 cm. The

data are as fo11ows

j 123456 7 8 9 10 11 12 13 14

(3.2) 1 5 20 38 50 110 110 104 66 44 18 10 1 1

Splines and Histograms 293

B.K.S. show in [3, 13, Figure 2] the graph of the

cardinal quadratic histospline fitted to the data (3.2).

Their curve shows four relative extremes (polymodality).

Our Figure 2 shows the histogram H = (h.) and the J

corresponding quartic histospline 04(x) of Theorem 2

is (of the two curves shown) the one that is also

endowed with 4 extreme points. In fact our curve and

B.K.S.'s are not very different.

At this point the author decided to try some

further smoothing by the simple device of doubling

the length of all cZass intervals. In this way we

obtain 7 class intervals, all of length 2, and (3.2)

gives the new histogram

described by the table

j 1 2 3

(3.3)

* H

4 5

(j = 1, ... , 7)

6 7

* h. 6 58 160 214 110 28 2 J

Notice that the heights of the new rectangles are

* * h./2 • J

The corresponding histospline 04(x) was

computed by splint and its graph is also shown in

* Figure 2, as weIl as H * The appearance of 04(x)

294 1.J. Schoenberg

seems to indicate that the doub1ing of the c1ass

intervals was just about the correct amount of smoothing

that the data (3.2) required. Even so, we did not quite

* escape the b1emish of negative va1ues of 04(x) . Whi1e

* 04(3.4) = 113.312 is c10se to the maximal va1ue, we

found that

x 6.5 6.6 6.8 7.0

.344 -.085 -.055 .000 * ° 4 (x)

* We fina11y remark that the graph of 04(x) is rather

c10se to the curve obtained by I.J. Good (see [3, 54,

Figure 6]) for the data (3.2) by an entire1y different

approach.

4. A SECOND CONSTRUCTION OF 04(x) .

Let

(4.1) {cr (x) }

denote the subspace of S4[O,m] of those elements

o(x) that satisfy the B.C.

Splines and Histograms 295

Figure 2

" L - - - - - ~"',~--~--t--------f

o o - o r-

....

296 I.J. Schoenberg

(4.2) a(O) = a'(O) = 0, a(m) = a'(m) = 0 .

The main idea is to work within this subspace (4.1) and

to find a convenient basis for it.

A basis meeting all requirements is furnished by

I7, Theorem S, 81-82] as fo110ws. The general4th

degree B-sp1ine is obtained from

(4.3)

in the form

M(x;t) 4 = 5(t - x)+

This is the fifth-order divided difference of (4.3),

with respect to the variable t, and based on the

six points

(4.5)

x. 1

satisfying the conditions

The function (4.4) is defined for all real x, it is

positive in (xo'xS) and vanished everywhere in its

comp1ement. M(x) E C3(R) as 10ng as the knots (4.5)

are all distinct. However, if x. 1

4-v mu1 tiplicity v, then M(x) E C

is a knot of

near Finally,

Splines and Histograms

M(x) is so normalized that

00

(4.6) J M(x)dx = 1 . _00

For simplicity, let us assume that

(4.7) m > = 5 .

From [7, Theorem 5] it follows that we obtain a

basis of (4.1) as folIows: We consider the knots

x = ° and x = m as tripZe knots and write accor­

dingly

(4.8) ° , ° , ° , 1, 2, . . • , m-l , m , m , m •

We group these m + 5 elements in all possible ways

into 6 aonseautive ones (call one such group

xo ,xl , ••• ,x5) and form the corresponding B-splines

(4.4). The m B-splines so obtained form a basis of

* S4[O,m] • We repeat this statement as

LEMMA 3. If UJe UJY'ite

~l(x) M(x;O,O,O,1,2,3)

~2(x) = M(x;O,O,1,2,3,4)

297

298 I.J. Schoenberg

(4.9)

~3(x) = M(x;0,1,2,3,4,5)

~4(x) = M(x;1,2,3,4,5,6)

~m_2(x) = M(x;m-5,m-4,m-3,m-2,m-1,m) ,

~m_1(x) = M(x;m-4,m-3,m-2,m-1,m,m) ,

~ (x) = M(x;m-3,m-2,m-1,m,m,m) , m

then these m funations form a basis for the spaae

defined by (4.1), (4.2).

The notations used in (4.9) assumed (4.7). If

m < 5 then obvious changes are necessary. Thus in the

extreme case that m = 1 , then the basis (4.9) reduces

to the single element

122 = M(x;O,O,O,l,l,l) = 30 x (1 - x) in [0,1] •

Every element of (4.1) having a unique represen­

tation

(4.10) m

a(x) = L y~~~(x) 1

we may now inforce the area-matching property

Sp1ines and Histograms

i (4.11) f cr(x)dx = h.

i-I ~

as folIows. Writing

(4.12) A. ~a

i

= f i-I

(i=l, ... ,m)

4> (x)dx a

and substituting (4.10) into (4.11) we obtain the

relations

(4.13) h. ~

(i = 1, ... ,m) •

299

This is a non-singular system since it defines cr(x)

unique1y. Due to the structure of the functions (4.9),

the system (4.13) has a 5-diagona1 matrix.

Solving the system (4.13), we obtain the Ya and

then cr4 (x) = cr(x) is expressed by (4.10). To use this,

we need to know the 4> (x). These are given by the a

fo110wing exp1icit formu1ae:

(4.14) 5 4 27 4

4> (x) = -- {(3 - x) - -- (2 - x) + 27(1 1 54 + 4 +

if x ~ 0 ,

300

(4.15)

I.J. Schoenberg

~ ( ) = ~ {(4 - x)4 _ 16 (3 _ x)4 '+'2 x 96 + 3 +

4 4 + 12(2 - x)+ - 16(1 - x)+}

if x ~ 0 ,

whi1e both these functions are o if x < 0 .

Moreover

(4.16) (-00 < X < 00) ,

~4(x) ~3(x - 1)

~5(x) ~3(x - 2)

(4.17)

~m_2(x) ~3(x - m + 5) ,

and fina11y by symmetry

(4.18) ~ (x) = ~ (m -m-1 2 x) , ~ (x) ~1 (m - x) . m

The elements (4.12) of the matrix IIA. 11 are 1(1.

obtained from (4.12) and (4.14) - (4.18) by direct

Splines and Histograms 301

integration. These elements are rational numbers.

Reducing them to their least common denominator, which

is 4320 = 25335 we obtain that

(4.19) 4320 IIA.all = 1

2300 525 36 o o

1940 2595 936 36

80 1155 2376

o 45 936

o 0 36

o o

45 0

1155 80

2595 1940

o 525 2300

This matrix is symmetrie with respect to its center.

The elements of the columns 3,4, •.• ,~2, are all

identical with the numbers

(4.20) 36 xl, 36 x 26 , 36 x 66 ,

36 x 26, 36 xl,

302 I.J. Schoenberg

appropriate1y shifted. Thus, if m = 5 , then (4.13)

becomes

2300 525 36 0 0 Y1 hJ.

1940 2595 936 45 0 Y2 h2

80 1155 2376 1155 80 Y3 = 4320 h3

0 45 936 2595 1940 Y4 h4

0 0 36 525 2300 Y5 h5

We may use (4.19) even if m = 4 , when the co1umns

with elements (4.20) are missing a1together.

11. TRE BIVARIATE GASE

5. TRE MAIN RESULTS.

Para11e1ing the deve10pments of §1 we discuss

the approximation of bivariate density functions by

means of biquadratic and biquartic sp1ine functions.

Let us exp1ain the problem and our termino1ogy.

Let

(5.1) R = [O,m] x [O,n]

Splines and Histograms 303

be a rectangle in the (x,y)-plane of dimensions m

and n, where m and n are natural numbers. We

think of it as dissected into mn unit squares

(5.2) Rij = Ii-l,j] x Ij-l,j] •

A bivapiate histogpam

(5.3) H = (hij ), (i = l, ••. ,m; j = l, ••• ,n) ,

is a matrix of observed frequencies, where h.. is 1J

the number of observations falling within the square

Rij •

DEFINITION 2. An integpabZe funation f(x,y) ~ defined

in R ~ is said to possess the voZume-matahing ppopepty

with pespeat to H ~ ppovided that

(5.4) 11 f(x,y)dxdy = hij for all (i,j). Rij

We denote by the syrriboZ VM(H) the aZass of funations

that matah the mn voZumes of H •

DEFINITION 3. Let k be a natuPaZ nurribep. We denote

by

(5.5) Sk,k(R) = {S(x,y)}

304 1.J. Schoenberg

the class of functions S (x,y) " defined in R" and

having the following properties:

(5.6) In each Rij , S (x,y) is of the form

k k a(i,j)xayß S (x,y) = L L

11.=0 ß=o aß '

and hence depends on (k + 1)2 parameters.

(5.7) The k2 partial derivatives

are continuous in R.

Our main resu1ts are the fo110wing two theorems.

THEOREM 3.

1. There is a unique G2(x,y) such that

(5.8)

and satisfying the B.C.

(5.9)

along the entire boundary of R.

Sp1ines and Histograms

2. Among aZZ functions f(x,y), defined in R, and

satisfying the three conditions:

(i) f (x,y) E VM(H) ,

(5.10) (ii) f(x,y) = 0 if (x,Y) E aR ,

305

(iii) f(x,y) is absoZuteZy continuous in the

sense of [5, 653],

the biquadPatic spZine cr 2 (x,y) has the optimaZ

property

(5.11) f f (f (x,y»2dxdy > f f (cr2 (x,y»2dxdy , R xy R ,xy

unZess f(x,y) = cr(x,y) throughout R.

THEOREM 4.

1. There is a unique cr 4(x,y) such that

(5.12)

and satisfying the B. C.

(5.13) cr4 (x,y) = cr 4 (x,y) = cr4 (x,y) = 0 ,x ,y

if (x,y) E aR •

306 1. J. Schoenberg

2. Among all functions f(x,y), defined in R, and

satisfying the thpee conditions:

(5.14)

(i) f(x,y) E VM(H) ,

(ii) f(x,y) = f (x,y) x

= f (x,y) = 0 y

if (x,y) E aR ,

(iii) The mixed paPtial depivative f (x,y) xy

absolutely continuous in the sense of

[5, 653],

is

the biquaptic sp Une °4 (x, y) has the op tima l ppopepty

(5.15) f f (f (x,y»2dxdy > f f (04 (x,y»2dxdy , R xxyy R' xxyy

unless f(x,y) = 0 4 (X,y) thpoughout R.

According to [5, 654, Satz 1 und Satz 2] f(x,y) -

is abso1ute1y continuous in R if and on1y if it

admits in R a representation

x y x y f(x,y) = f f g(x,y)dxdy + f gl (x)dx + f g2(y)dy + C ,

o 0 0 0

where g(x,y), gl (x) , and g2(x) , are summab1e

functions.

Sp1ines and Ristograms 307

6. ON TRE REPRESENTATION OF BIVARIATE SPLINES.

In the proofs that are to fo110w we are going to

concentrate main1y on the biquartic case of Theorem 4.

Let

(6.1) * S4,4(R) = {o(x,y)}

denote the subspace of S4,4(R) of those elements

o(x,y) that satisfy the B.C.

(6.2) o(x,y) = 0 (x y) = 0 (x,y) = ° if (x,y) € eR , x' y

our immediate aim being to state an ana10gue of Lemma

3 of §4. For this purpose we need the m B-sp1ines

(4.9) and also n further B-sp1ines, functions of

y , defined by

(6.3)

1/I1 (Y) = M(y;0,0,0,1,2,3)

1/I2(Y) = M(y;0,0,1,2,3,4)

1/I3(Y) = M(y;0,1,2,3,4,S)

1/In-2(y) = M(y;n-S,n-4,n-3,n-2,n-1,n)

1/In-1(y) = M(y;n-4,n-3,n-2,n-1,n,n)

1/In (Y) = M(y;n-3,n-2,n-1,n,n,n) •

308 I.J. Schoenberg

* LEMMA 4. The spaae S4,4(R) has 4the dimension mn

and

* (6.4) a (x, y) € S 4 , 4 (R)

impZies a unique representation in R of the form

(6.5)

where

a(x,y) = m n I I y .. 4>. (x)1/Jj (y)

i=l j=l 1J 1

are appropriate aonstants.

* That (6.5) furnishes on1y elements of S4,4(R)

is c1ear, because all functions 4>i(x)~j(Y) be10ng to

this space. To save space we omit the e1ementary but

long proof of the converse statement.

Remark: earl de Boor adds the fo11owing interesting

observations: 1. The first part of Theorem 3 fo11ows

from de Boor's 1962 resu1t concerning bicubic sp1ine

interpolation (for reference see [1, 278]). 2. On

the basis of Lemma 4 the first part of Theorem 4

fo11ows from Theorem 2 by an appeal to general pro­

perties of the tensor product. For details see

de Boor's Appendix to the present paper.

Sp1ines and Histograms

7. PROOFS OF THE FIRST PARTS OF THEOREUS 3 AND 4.

PROOF FOR THEOREM 4: In terms of the histogram (5.3)

we define the cumu1ative matrix

(7.1) F = (F ij) (i = 0, ... ,m; j 0, ... ,n)

by setting

(7.2) F. = F . 1.,0 O,J

= 0 F .. = 1.,J

if i > 0, j > 0

In terms of cr(x,y) , defined by (6.5), we define

x y (7.3) S(x,y) = J J cr(u,v)dudv, (x,y) ER.

o 0

This is a biquintic sp1ine function, and (7.3) shows

that

(7.4) cr(x,y) = S (x,y). xy

From (7.3), (7.4), and (6.2), we see that S(x,y)

satisfies the B.C.

(7.5) S(x,O) = 0, S(O,y) = 0

if 0 < x ~ m, 0 ~ y < n ,

309

310 I.J. Schoenberg

(7.6) S (x,y) = S (x,y) = S (x,y) = 0 xy xxy xyy

if (x,y) € aR •

Finally, from (7.3) and (6.5), we find that

m n x y (7.7) S(x,y) = I I Yij (! ~i(u)du)(! wj(v)dv)

i=l j=l 0 0

and that it depends also on mn parameters.

LEMMA 5. The paroameteps Y ij can be uniquely detep­

mined such that

(7.8) S(i,j) = Fij , (I! i ~ m, 1 ~ j ~ n) .

PROOF OF LEMMA 5. It suffices to show that if

(7.9) S(i,j) = 0 fop alZ (i,j)

then

(7.10) Yij = 0 fop alZ (i,j).

Assuming (7.9) to hold, we conclude that

Sp1ines and Histograms

X \I (7.11) S(x,\I) = ~ lj Yij (! ~i(u)du)(J ~j(v)dv)

1 0 0

(\I fixed integer)

is a quinticsp1ine vanishing if x = O,l, ..• ,m .

Moreover, its first and second derivatives vanish

at both ends: From (7.11)

(7.12) Sx (x, \I) ~. (v)dv , J

\I (7.13) Sxx(x,\I) = L L Yij~~(x) J ~j(v)dv,

o

and both vanish if x = 0 , or x = m , because all

311

~i(x) have the property that ~i (0) = ~! (0) = ~i(m) = 1

~i(m) = 0 . We conc1ude from Lenuna 2 ( §l) that

S (x, \I) = 0 if o < x < m •

Therefore also S (x,\I) = 0 vanishes identica11y, for x

each \I . Now (7.12) imp1ies that

m n \I

L ~i(x) L Y .. J ~. (v)dv = 0 if 0 < x < m , j=l 1J J = i=l 0

312 1.J. Schoenberg

and from the linear independence of the ~i(x) we

conc1ude that

n \I

(7.14 ) L Yij J n=l 0

l/Jo (v)dv = 0 ~

for each

Keeping i fixed we observe that

(7.15) I YiJo JY l/JJo(v)dv

j=l 0

i and each \I •

is a quintic sp1ine in Y, vanishing for Y = O, ... ,m

by (7.14), and having the first two derivatives

which also vanish if y = 0 or y = n 0 Again by

Lemma 2 we conc1ude that the quintic sp1ine (7.15)

vanishes identica11y. Therefore also its derivative

vanishes identica11y so that

if o < y < n •

From the linear independence of the l/Jj(Y)

that (7.10) indeed ho1ds.

we see

Sp1ines and Histograms 313

We return to the proof of Theorem 4. By Lemma 5

we know that (7.8) are satisfied by a unique S(x,y) •

It is now easy to show that the quartic sp1ine a(x,y) ,

defined by (6.5), or equiva1ent1y, by (7.4), satisfies

all the conditions of the first part of Theorem 4.

Since * a(x,y) E S4,4(R) we know that the B.C. (5.13)

are satisfied. We are yet to show that

(7.16) a(x,y) E VM(H) •

To show this we use (7.3), or (7.4), and observe that

JJR a(x,y)dxdy = S(i,j) - S(i-1,j) - S(i,j-1) ij

+ S(i-1,j-1)

F .. - F. 1 . - F. j 1 + F. 1 . 1 1J 1-, J 1, - 1- , J -

= h .. 1J

in view of the definition (7.2) of the F ..• 1J

A proof of the first part of Theorem 3 is entire1y

simi1ar and not any shorter. It is based on ana10gues

of Lemmas 3 and 4 for quadratic and biquadratic sp1ines,

respective1y. The ro1e p1ayed by Lemma 2 is taken over

by Lemma 1 on cubic sp1ine interpolation. Fina11y, the

314

partial derivatives o , x

1.J. Schoenberg

o ,do not appear and there­y

fore the B.C. (7.6) are to be rep1aced by

S (x,y) = 0 if (x,y) E aR . xy

8. PROOFS OF THE SECOND PARTS OF THEOREMS 3 AND 4.

PROOF OF THEOREM 3. Let 02(x,y) = o(x,y) be the

biquadratic sp1ine of the first part of Theorem 3,

and let f(x,y) satisfy the three conditions (5.10).

We first note the identity

JJ (f -0 )2dxdy R xy xy

2 JJ (0 ) dxdy R xy

- 2JJ (f -0 )0 dxdy R xy xy xy

and wish to show that the last integral vanishes.

In view of (5.9), and (5.10)(ii), we see that the

difference

(8.2) ~(x,y) = f(x,y) - o(x,y)

satisfies the B.C.

Sp1ines and Histograms

(8.3) ~(x,y) = 0 if (x,y) E aR •

In any case we may write

ff (f - er )er dxdy RXY xyxy

n m

ff ~ er dxdy R xy xy

n m = / {f ~ er dx}dy =

o 0 xy xy / {j er d ~ }dy o oxyxy

315

However, from (8.3) we conc1ude that ~ = 0 on the two y

vertica1 sides of dR. On integrating by parts the

last inside integral, we therefore find the last

repeated integral to be

n m -/ {J ~ er dx}dy

o 0 y xxy

m n -J {J ~ er dy}dx yxxx

o 0

and by (8.3) this is

m n = J {J ~er dy}dx

o 0 xxyy

We have just shwon that

-J J ~ er dxdy R y xxy

m n -J {j er d ~ J dx

o 0 xxy y

JJ ~er dxdy. R xxyy

316 I.J. Schoenberg

(8.4) ff (f - 0 )0 dxdy RXY xyxy

1f (f - 0)0 dxdy. R xxyy

Observe thet 0 (x,y) is a step-function that has a xxyy

constant value inside eaah square We

may therefore write the last integral as

L c.j(ff fdxdy - JJ odxdy) i,j 1 R. . R ..

1J 1J

and all terms of this sum clearly vanish because f as

weIl as 0 belong to VM(H) •

Therefore (8.1) reduces to

J J (f ) 2dxdy = R xy

However, the last integral vanishes if and only if

f = 0 holds almost everywhere in R, so that xy xy

everywhere

f (x, y) x y

J J f (u,v)dudv o 0 xy

for all (x,y) ER.

x y J J 0 (u,v)dudv = o(x,y) , o 0 xy

Splines and Histograms 317

PROOF OF THEOREM 4. The proof is very similar and only

slightly more elaborate. Let a4(x,y) = a(x,y) be

the biquartic spline of the first part of Theorem 4.

It satisfies the B.C.

(8.5) a = a = a = 0 along 3R , x Y

while the Itarbitrarylt function f(x,y) satisfies the

simila r B. C.

(8.6) f = f = f = 0 along dR. x y

In order to establish the inequality (5.15) we write

an identity similar to (8.1), with f , xy

a xy replaced

by f xxyy' a respectively. xxyy , Again we wish to

show that its last integral vanishes. Writing

(8.7) ~(x,y) = f(x,y) - a(x,y) ,

this amounts to showing that

(8.8)

We first write

318 1. J. Schoenberg

n m ff ~ cr dxdy

R xxyy xxyy f {f ~ cr dx}dy

o 0 xxyy xxyy

n m f If cr d ~ }dy o 0 xxyy x xyy

However, ~ = f x x

cr = 0 a10ng aR and therefore x

,j, - 0 '+'xyy

a10ng the two vertiaat sides of aR.

The last repeated integral is therefore

n m -f {f cp cr dx}dy =

o 0 xyy xxxyy -ff cp cr dxdy xyy xxxyy

R

m n m n -f {f cp cr dy}dx xyy xxxyy

o 0

-f {f cr d cp }dx xxxyy Y xy o 0

Since a10ng aR we conc1ude that

o

a10ng the two horizontat sides of aR.

Sp1ines and Histograms

We may therefore integrate the last integral by parts

and find it to be

m n = f {f ~ cr dy}dx =

o 0 xy xxxyyy ff ~ cr dxdy •

R xy xxxyyy

We repeat the entire operation once more as fo110ws:

n m

319

= -J {J ~ cr dx}dy = o 0 y xXXXYYY

-Jf ~ cr dxdy y xxxxyyy R

m n m n = -f {f cr d ~}dx

o 0 xxxxyyy y = f {f ~cr dy}dx

o 0 xxxxyyyy

= ff ~cr dxdy • xxxxyyyy R

Observe that the last integration by parts was 1egiti-

mate even though cr xxxxyyy is not a continuous functio~

The reason: The integration by parts was performed with

respect to y, for a fixed vaZue of x , so that

cr is a aontinuous funation of y • xxxxyyy

Fina11y the last integral vanishes for the same

320 1. J. Schoenberg

reason as in the proof of Theorem 3: a is a xxxxyyyy

step-function, while ~ matches the volumes of the

"zero"-histogram. Hence (8.8) is established and the

analogue of (8.1), that we started from, establishes

the inequality (5.15), possibly with the equality sign.

In fact we do have equality in (5.15) if and only if

(8.9) f (x,y) = a (x,y) xxyy xxyy

almost everywhere in R.

If we integrate both sides of (8.9) over the rectangle

[O,x] x [O,y] we find that

f (x,y) - f (x,O) - f (O,y) + f (0,0) xy xy xy xy

(8.10) = a (x y) - a (x 0) - a (O,y) + a (0,0) xy' xy' xy xy

if (x,y) ER.

On the other hand, from (8.5) and (8.6) we find, by

partial differentiation, that f and a vanish xy xy

on aR, so that (8.10) reduces to

f (x,y) = a (x,y) in R. xy xy

Splines and Histograms 321

One more integration of both sides over IO,x] x IO,y]

shows that indeed f(x,y) = o(x,y) everywhere in R.

9. A CONSTRUCTION OF THE BIQUARTIC HISTOSPLINE 04(x,y).

Not only did the B-spline representation of Lemma

4 furnish the foundation of a proof of the first part

of Theorem 4, but it also allows to solve conveniently

the numerical problem involved, as folIows. We know by

lemma 4 that

(9.1) o(x,y)

represents the most general biquartic spline satisfying

the B.C. (5.13). In order to satisfy also the volume-

matching conditions

(9.2) ffR o(x,y)dxdy ij

h .. 1J

We substitute the expression (9.1) into (9.2). Using,

as in (4.12), the symbols

(9.3) i

f ~a(x)dx i-I

j J 1)JS(y)dy j-l

322 I.J. Schoenberg

we obtain the system

m n (9.4) I I A. BjßY ß = h .. for all (i,j).

a=l ß=l ~a a ~J

This, then, is the linear system of mn equations in as

many unknowns Yaß that has to be solved.

In (4.19) we have given the numerical values of the

Aia The matrix IIBj ß 11 has the same structure and in

fact the same elements appear in both, since

All = Bll = 2300/4320, A12 = B12 = 525/4320 a.s.f.

Of course, they differ in their sizes.

How sparse is the matrix of the system (9.4) ?

This question is easily answered: The matrix IIAia 11

has, by (4.19), exactly m + 2(m-l) + 2(m-2) = Sm - 6

positive elements, and similarly IIBjßl1 has Sn - 6

positive elements. It follows that the number of non­

vanishing (actually positive) coefficients of the

system (9.4) is exactly equal to

(9.5) (Sm - 6)(5n - 6) •

Sp1ines and Histograms

2 2 This is considerab1y sma11er than mn x mn - m n

especia11y if m and n are not sma11. Thus if

323

m = n = 7 we find that among the 49 2 = 2401 possib1e

elements of the matrix of (9.4) exact1y 841, or 35%,

are positive.

Mrs. J. Gray, who wrote the program Splint of §2,

will hopefu11y soon also write a computer program for

the solution of the system (9.4).

Dur last comment concerns the choice of the

coordinate axes. This requires some exp1anations.

Suppose that we are given a scatter diagram (S.D.),

or set of observed points, indicated in Figure 3 by an

ova1-shaped figure. Let xOy be the original

coordinate axes. In the case of Figure 3 it wou1d not

be advisab1e to work within a rectang1e.

R = {O ~ x < m, 0 < y < n} ,

for many of the ce11s R .• 1J

wou1d carry vanishing

frequencies h.. . 1J

Dver all such ce11s the histosp1ine

G4 (X,y) wou1d assume sma11 va1ues of both signs and

to no good purpose. It seems reasonab1e in this case

to change to the new coordinate system x'o'y' and to

324

enclose the S.D. within a rectangle

I.J. Schoenberg

R' = {O ~ x' < m' = '

o ~ y' ~ n'} that would clearly show fewer cells with

vanishing frequencies. If convenient, the frame

x'o'y' need not even be orthogonal, since all results

remain valid for oblique axes. This approach is clearly

only then possible (expecting translation ofaxes) when

all the original observations are available and not

merely a histogram.

Splines and Histograms 325

( ?n,', ",,' )

o

Figure 3 •

326 I.J. Sehoenberg

REFERENCES

1. Ah1berg, J.H., Nilson, E.N. and Wa1sh, J.L.: The theory of sp1ines and their app1ieations, Aeademie Press, New York/London, 1967.

2. B1iss, C.I.: Statist1es in Bio1ogy, Vo1. 1, Me Graw-Hi 11 , New York, 1967.

3. Boneva, L.I., Kenda11, D.G. and Stefanov, I.: Sp1ine transformations: Three new diagnostie aids for the statistiea1 data-ana1yst, J. of the Royal Statistiea1 Soe., Series B, 33 (1971), 1-70.

4. de Boor, C.: On ea1eu1ating with B-sp1ines, J. of Approximation Theory, 6 (1972), 50-62.

5. Caratheodory, C.: Vorlesungen über reelle Funktionen, Seeond Edition, B.G. Teubner, Leipzig-Ber1in, 1927.

6. Car1son, R.E. and Hall, C.A.: On pieeewise po1ynomia1 interpolation in rectangular polygons, J. of Approx. Theory, 4 (1971), 37-53.

7. Curry, H.B. and Sehoenberg, I.J.: On Po1ya frequeney funetions IV. The fundamental sp1ine funetions and their limits. J. d'Ana1yse Math. (Jerusa1em), 17 (1966), 71-107.

8. Grevi11e, T.N.E.: Introduetion to sp1ine funetions, 1-35 in Theory and app1ieations to sp1ine funetions (T.N.E. Grevi11e, Ed.), Aeademie Press, New York/ London, 1969.

Sp1ines and Histograms

9. Schoenberg, I.J.: Contributions to the problem of approximation of equidistant data by ana1ytic functions, Quart. App1. Math. 4 (1946), 45-99, 112-141.

10. Schoenberg,I.J.: Notes on sp1ine functions 11. On the smoothing of histograms, MRC Tech. Sumo Report #1222, March 1972, Madison, Wisconsin.

327

, , APPENDIX TO SPLINES AND HISTOGRAMS BY I.J. SCHOENBERG

Car1 de Boor

It is the purpose of this appendix to point out

the tensor product structure of the construction in

part 11, thus re1ating it to known resu1ts concerning

interpolation by tensor product sp1ines. The addi­

tional a1gebraic machinery required seems worth

knowing since one may use it to advantage for the

efficient ca1cu1ation of bivariate histosp1ines.

1. ODD DEGREE SPLINE INTERPOLATION AND RELATED

INTERPOLATION PROBLEMS.

329

We begin with a recapitu1ation of known resu1ts

(see e.g., I8]) concerning odd degree interpolation on

arbitrary meshes.

330 C. de Boor

Let

m ~ = (x.)

1 0

be a partition for the interval Ia,b] , i.e.,

We denot~by Sk(~) the class of spline functions of

degree k, defined on Ia,b] and having the points

x1"",xm_1 as simpZe knots. This means that Sk(~)

is contained in cCk- l ) [a,b] and that the restrietion

of each sex) E Sk(~) to the interval (xj_l,Xj ) is

a polynomial of degree not exceeding k, (j = l, ••• ,m).

Sk(~) is a linear space of dimension k + m , and,

according to [7, Theorem 5], a basis for Sk(~) is

given by the sequence

of B-splines, with

and

k = (k + l)(t - x)+

Appendix to Splines and Histograms

f a , for i < 0

xi -1 b , for i > 0 .

LEMMA Al. POP given f € C(k)Ia,b] 3 thepe exists

exactty one s € S2k+l(~) such that

331

s(a) f(a), SI (a) = f' (a), ..• ,s (k) (a) = f(k) (a) ,

s (x.) = f (x.), i = 1, ... , m-l 1. 1.

s (b) = f (b) s'(b) = f'(b), .•• ,s(k)(b)

Denote this s by P f Then ~

a a

top att g € C(k)[a,b] with absotutety continuous k-th

and squaPe-integpabte (k+l)st depivative othep than

g = P f top which P g = P f . ~ ~ ~

Now observe that two functions fand g agree

at xo, .•. ,xm if and only if fand g agree at one

of these points, say at x a, and o

332

f(x.) - fex. 1) = g(x.) - g(x. 1)' i ~ ~- ~ ~-

If f is abso1ute1y continuous, then

hence

x fex) f(a) + ! f' (y)dy

f(x.) - fex. 1) ~ ~- f

a

x. ~

f' (y)dy

C. de Boor

1, ... ,m •

Two sbso1ute1y continuous functions fand g agree,

therefore, at xo, ... ,xm if and on1y if

x. ~

x. ~

f(a) g(a);! f' (y)dy f g' (y)dy i 1, .. . ,m .

Since

Lemma Al has therefore the fo11owing

CORROLLARY. POP evepy

exaat Zy one

f s C(k-1) [a,b] ~ thepe exists

~ denoted by p (1) f ~ fop UJhiah 'IT

Appendix to Splines and Histograms

X. l.

J Xi

s(x)dx = J f (x) dx

333

j = 0, ... , k-I ,

i=I, ... ,m.

Furthep, odd-degpee spZine inteppoZation is peZated to

this apppoximation scheme by

(P (1) f) (x) Tr

x = (p J f(y)dy) (1) (x)

Tr a

Schoenberg's Theorems 1 and 2 and their corollaries

follow from this. More generally, one has

THOEREM Al.

Zet AM(H)

m Fop a given peaZ sequence H = (hi)l '

denote the cZass of functions

g E c(k-l)[a,b] satisfying

(i) g(k-l) is absoZuteZy continuous and

g(k) E L2 [a,b]

334

(H)

X. 1

f X. 1 1-

C. de Boor

g(x)dx = h. 1

i = 1, ... ,m

(Hi) g(1) (a) = gO) (b) = 0, j = 0, ..• ,k-1 •

Then, S2k(~) and AM(H) have exaatly one element in

aommon, denoted by

A (H) , ~

and this element A (H) also uniquely minimizes ~

b (k) 2 f [g (x)] dx a

over g E AM(H) •

In order to compute A (H) , consider the basis ~

for S2k(~)' One checks that

~(j)(a) ~ 0 if and on1y if i

~~j)(b) ~ 0 if and on1y if 1

i - j-k+1l, i = m+k-jJ

j = 0, ... ,k-1

Appendix to Splines and Histograms

Hence, after leaving off ~l-k'···'~o and

~ nrl-l' ... , P nrl-k , the remaining sequence

~l'···'~m

is a basis for

* {sES 2k (1T) IsO) (a) s (j) (b) S 2k (1T) = 0,

Let now A be the m x m matrix given by

x. ~

j

A(i,j) J ~. (x)dx , J

i,j l, ... ,m. x i - l

Then

A (H) 1T

m L Y.~.

. 1 ~ ~ ~=

335

O, ••• ,k-l}.

with r (Yi ) the solution of the linear system

Ar = H .

Before deriving the corresponding results for

bivariate splines, we introduce some language concerning

linear interpolation problems, which we have found

336 C. de Boor

convenient in the discussion of tensor product schemes.

If ~l""'~m is a sequence in the linear space

F , and Al" .. ,An is a sequence of linear functionals

on F, then we may consider the Linear InterpoZation

ProbZem (LIP) given by (A.) : to find, for J

given f E F , a linear combination g = L.a.~. 111

of

the ~. 's so that 1

A.g 1

A.f 1

i l, ... ,n.

Actually, the problem does not depend on the

individual

span

cfJ i 's and A. 's, but only on the linear J

of ~. 's and the linear span 1

of the Ai 'so This is quite clear for the ~i 's,

since "a linear combination g = L.a.~. of the ~1' 's" 1 1 1

Appendix to Splines and Histograms 337

is equivalent to "a g E: Fl " which makes no reference

to the individual Pi 'so But, also,

if and only if

A.g A.f i = l, ... ,n 1. 1.

E.a.A.g 1. 1. 1.

Hence, the LIP can also be stated: to find, for

given f E: F ,a g E: Fl so that

Ag Af, for all A E: A •

We say tht the LIP given by Fl and A is

correct if it has exactly one solution for every f E: F.

The following lemma is not difficult to prove:

LEMMA A2. Let

(Aj)~ be a basis for A. Then

Ci) the LIP given by Fl and A is correct if and

only if the Gramian matrix

338 c. de Boor

is invertibZe; in partiauZar~ n = m is a neaessary

(but not a suffiaient) aondition for the aorreatness

of the LIP.

(ii) if the LIP given by FI and A is aorreat~

then the interpoZant Pf for given f E F aan be

aomputed as

Pf with

In these terms, Lemma Al states that the LIP

given by FI = S2k+I(n) and

A2k+l , (k) (1) (k)

= [8 , ... ,8 ,8 , ... ,8 ,8b ,8b , ... ,8b ] a a xl xm_l

is correct, where, by definition, is the linear

functionalon c(k)[a,b] given by the rule

and 8 is short for 8(0) a a

Appendix to Splines and Histograms

2. TENSOR PRODUCTS OF LINEAR SPACES OF FUNCTIONS.

The tensor product of two (or more) algebraic

structures is a weIl understood construct of Algebra.

But since we only need a few notions concerning the

tensor product of two linear spaces of functions, we

give a short discussion of this special case in order

to spare the reader an excursion into an abstract

algebra text.

Let F be alinear space of functions, all

defined on some set X into the reals, and let G

be, similarly, a linear space of functions defined on

some set Y into R. For each f E Fand each

g E G , the rule

h(x,y) = f(x)g(y), all (x,y) E X x Y

defines a function on X x Y , called the tensor

produat of f with g and denoted by

f ® g •

Further, the set of all finite linear combinations of

functions on X x Y of the form f ® g for some

f E Fand some g E G is called the tensor product

of F with G and is denoted by F ® G. Thus,

339

340 C. de Boor

n F ® G = { La. f. ® g. la. E: R, f. E: F, g. E: G,

i=l ~ ~ ~ ~ ~ ~

i = 1, .•. , n; some n}

and F ® G is a linear space.

A simple and important example is provided by

polynomials in two variables. Taking F Ph ' the

linear space of polynomials of degree < h , as

functions on X = R and similarly G = P as k

functions on Y = R we easily recognize F ® G as

the linear space Ph,k of all polynomials in two

variables of degree < h in the first and of degree

< k in the second variable, considered as functions

on the plane 2 xxy=R.

A second simple example arises with the choice

Rm F = , the linear space of real m-vectors considered

as functions on X = {1,2, •.. ,m} , and, similarly,

Rn G considered as a linear space of functions on

y {l,2, ... ,n} In this case, F ® G is the linear

space of all m x n matrices, considered as functions

on X x Y = {(i,j) li = l, ... ,m; j = l, ... ,n}

Appendix to Sp1ines and Histograms 341

One verifies that the tensor product is biZinear,

i.e., the map

F x G + F ® G : (f,g) I + f ® g

is linear in each argument:

In particu1ar,

F ® G = {E.f. ® g. If. E F, g. E G, i ~ ~ ~ ~ ~

1, ••• ,n; some n}

which saves a 1itt1e writing.

Let now A and ~ be linear functionals on

Fand G, respective1y. One defines A ® ~ by the

ru1e

= E.(Afi)(~g·) , ~ ~

all E.f. ® gi. ~ ~

C1ear1y, if A ® ~ is a map on F ® G satisfying

(2.1), then A ® ~ is a linear functiona1 on F ® G •

But, (2.1) requires some discussion before we can

accept it as defining a map on F ® G. For, (2.1)

342 C. de Boor

makes use of the particu1ar form of Lifi ® gi ' i.e.,

the particu1ar f i 's and gi 's, to define A ® ~ on

Lifi ® gi. On the other hand, an element Z E F ® G

may be written in many different ways. 1f, e.g.,

Z = f ® g and f = f 1 + f 2 ,and g = 3g1 ' then

we can write Z as

f ® g or f 1 ® g + f 2 ® g or even as

Corresponding1y, the ru1e (2.1) wou1d give

(among others) for "the" va1ue of (A ® ~)z •

The doubts just raised can be dispe11ed as

fo11ows. 1f Z is any function X x Y , and y is

a particu1ar point in Y , then

Z (x) = z(x,y), all x E X Y

defines a function z on X, the y-seation of z • y

Appendix to Splines and Histograms

If, in particular,

Z = E.f. ® g. , 1. 1. 1.

for some f 's i

343

in Fand

gi 's in G,

then, by the definition of f. ® g. , we can compute 1. 1.

Z (x) as y

i.e.,

Z = Eig. (y)f. . y 1. 1.

This shows that Z E F , hence allows us to compute y

the number

(2.2)

Let now Z

function on

AZ ,and to compute it as y

AZ = E.g.(y)(Af.) . Y 1. 1. 1.

be the A-section of Z , Y defined by

ZA(y) AZ for all y E Y Y

i. e. , the

.

The notation is correct, ZA depends only on Z

A (and not on the particular f. 's and g. ' s) , 1. 1.

and

but

344 c. de Boor

can be computed by (2.2) as

z = E.(Af.)g. whenever z = E.f. ® g. 11111 1

This shows that ZA E: G , hence a110ws us to compute the

number ~ZA' and to compute it as

~Z~ = ~(E.(Af.)g.) = E.(Af.)(~g.) ~ 111 111

thus showing that E. (H . ) (~g. ) 111

depends only on A,~

and the function Z = E.f. ® g .• 111

We conc1ude that, for every linear functiona1

A on Fand every linear functiona1 ~ on G, (2.1)

defines a linear functiona1 on F ® G , and that this

functiona1 satisfies

(2.3) (A ® ~)z = A(Z~) = ~(ZA)' for all Z E: F ~ G •

, Here, Z is the ~-se()tion of Z ,

~

the resu1t of applying ~ to z(x,y)

of y for each fixed x E: X .

To give a simple examp1e, let F

and let

i. e. , Z (x) is ~

as a function

Appendix to Splines and Histograms

for some a and ß, and some integers rand

s < k , so that, e.g.,

Ai = f(r) (a) •

345

Then F ~ G is contained in C(k,k)(R2) , the space

of bivariate functions with k continuous derivatives

in each variable. Further (on F ~ G), A ~ ~ agrees

with the linear functional

v = ü(r,s) a,ß

since, for every f, g E c(k)(R) ,

v (f ~ g) f(x) g(y) I = x=a y=ß

With ~ changed to

y ~g = J g(y)dy

ß

f(r) (a)g(s) (ß)

= (H) Ülg) •

A ® ~ agrees with the linear functional v given by

the rule

346 C. de Boor

Y r r AZ = f (a lax )z(a,y)dy , 11 C(k,k)

a z E •

ß

THOEREM A2. Suppose that the GY'amian A = (A i 4>i) foY'

the sequenae 4>1, ••• ,4>m in F and the sequenae

A1, ••• ,Am of Zinear funationaZs on F is inveY'tibZe,

so that the LIP given by

is aOY'Y'eat. SimiZarZy, a8sume that B = (~i$j) is

inveY'tibZe, with $1, •. ·,$n E G and ~1'···'~n

Zinea:l' funationaZs on G, and set

FinaZZy, a8sume that (vij ) is a matY'ix (oY' doubZe

sequenae) of Zinear funationaZs on some ZineaY' spaae

H aontaining F ® G so that

Vijf ® g = (Aif)(~jg) , foY' aZZ i,j; atz

f ® g E F ® G .

Appendix to Spiines and Histograms

(ii) the LIP on H given by Fi ® Gi and

[(vij)ij] is aorreat, and

347

(iii) for given h EH, the interpoZant Rh aan

be aomputed as

(2.4) Rh = L. jr(i,j)~. x ~J' ~, ~

with

r = r = A-iL (BT)-i h h

where

Lh(i,j) = v .. h aU i,j . ~J

Remark. Here and beiow, we write D(i,j) rather than

Dij or dij for the (i,j)-th entry of a matrix D.

PROOF OF THE THEOREM. If h E Fi ® Gi ' then

(2.5) h = L . . r(i,j)~i ® ~. ~,J J

for some matrix r. But then

348 c. de Boor

In.(r,s) = Ar ~ ~ h = E. jf(i,j)(A $i)(~ $.) s 1, r s J

= (AfBT)(r,s) , all r,s

or

Since both A and Bare invertible by assunption,

and since ~ does not depend on the particular

representation (2.5) for h but only on h, this

implies the uniqueness of the expansion (2.5) for h,

therefore showing (i).

It follows further that, for a given matrix L

and a given h E F1 ~ GI ' we have ~ = L if and

only if the coefficient matrix f for H (with

respect to the basis ($i ~ $j)i,j of Fl ~ GI)

satisfies

proving (ii) and (iii);

The significance of (iii) for computations is

Appendix to Splines and Histograms 349

clear. Instead of having to solve the linear system

(2.6) L • • (A 4>.)(11 ljJj)r(i,j) = L (r,s) , ~,J r ~ s K

all r,s

of m x n equations in order to compute the coeffi­

cient matrix r for given h, one only needs to

solve two systems, of size m and n, respectively,

involving, respectively, n and m right sides.

More explicitly, having obtained by Gaussian

elimination a triangular factorization A A = A :t u

for A and B respectively in about

operations, one then computes

as

and

in about 2 2 O(n m + m n) operations. Straightforward

application of Gauss elimination would take

3 O«m x n» operations instead. The savings are

even more significant if (as in the applications below)

A and B are band matrices, a fact difficult to

exploit in a direct attack on (2.6) whatever the

actual ordering of r into a vector might be.

We close this section with aremark concerning

350 C. de Boor

the A.-sections and the llj-SectionS of the interpo-1.

lant Rh defined in the theorem. By (2.3) ,

A. (Rh) = (A. (1g 1l.)Rh A. (1g ll.h 1. ll. 1. J 1. J

J

while, as argued earlier, (Rh) E: FI since llj

Rh E: FI (1g GI . Hence, the ll.-section (Rh) J llj

Rh is the unique element s of FI for which

A.S 1.

i=l, ... ,m.

of

In words: The ll.-section of the interpolant to h is J

the interpolant to the ll.-section of h. J

This

establishes the

COROLLARY. Let h E: F I (1g GI . Then

h = 0 if and onZy if A. (1g ll.h = 0 for i 1, ... ,m. llj 1. 1.

SimiZarZy,

hA. = 0 if and onZy if A. ® ll.h 0 for j 1, ... ,n. 1. J 1.

Appendix to Sp1ines and Histograms 351

3. THE BIVARIATE CASE.

We consider the c1ass

C(k,k)(R)

of functions defined on the rectangle

R = [a,b] x [a' ,b']

and k times continuous1y differentiab1e in each

variable, i.e., having each of the (k + 1)2 partial

derivatives

continuous on R.

With ~ = (x.)m and 1. 0

i,j = 0, .•. ,k

n ~' = (y.) partitions

J 0

for [a,b] and [a',b'], respecitve1y, we define

as the c1ass of sp1ines of degree k (in each

variable) having the 1ines x = xi' i = 1, ••• ,m-1 ,

and j = 1, •.• ,n-1 , as simple mesh 1ines.

35Z c. de Boor

This means that Sk,k (7f x 7f') ~ C(k-1,k-1)(R) , and

that the restriction of each s e Sk k(7f x 7f') to , each of the mn rectang1e

is a po1ynomia1 of degree < k in each variable.

Even degree area matching sp1ine interpolation

to f e c(k-1)[a,b] by element of SZk(7f) (as

described in the coro11ary to Lemma Al) invo1ves

the Zk + m interpolation conditions

by

(3.1) A. f = ~

Xi

J f(x)dx l xi_1

f(i-m-1)(b) ,

m+k (A i )l-k given

i = 1, ... ,m

i > m

The corresponding scheme for SZk(7f') invo1ves

agreement at the Zk + n linear functiona1s

Appendix to Splines and Histograms

lll_k' •.• , II m+k given by the rules

f (-j) (a') j < 0 -

(3.2) lljf = jYj

f(y)dy j = 1, ... ,n y. 1 J-

f(j-n-l) (b ') j > n

The tensor product of these two schemes in the

spirit of Theorem A2 would then associate with each

f E C(k-l) [a,b] x C(k-l)[a',b'] the one element

s E S2k(TI) ® S2k(TI') for which

(A. ® 1l.)S ~ J

(A. ® ll.)f, i ~ J

l-k, •.. ,m+k

j l-k, ••. ,n+k

But this result is unsatisfactory for the reason

that we would prefer to interpolate by elements of

S 2k 2k (TI x TI') . ,

To overcome this objection, we prove that, for

all k,

353

354 C. de Boor

(3.3)

a special case of which is more or less the content

of Schoenberg's Lemma 4. For the proof, observe that

hence

= (k + m)(k + n)

so that (3.3) holds provided we can show that

(3.4) dim Sk, k ('If x 'If') ~ (k + m) (k + n) •

This we show by induction on k: For k = 0,

Sk k('If x 'If') consists of all functions on R which , are constant on each of the mn rectangles Rij ,

hence has dimension mn. Assuming (3.4) to hold for

given k = h , we make use of the fact that then

f E Sh+l,h+l('If X 'If') is absolutely continuous, hence

satisfies

Appendix to Splines and Histograms 355

x y f(x,y) = f(a,a') + f gl(r)dr + f g2(s)ds

a a'

x x + f f g(r,s)dsdr

a a'

with

g(x,y) = fxy(x,y) E Sh,h(~ x ~') •

Therefore

< 1 + h + m + h + n + (h+m)(h+n)

= (h+l+m)(h+l+n)

showing that (3.4) holds then for k = h + 1 , too.

This proves (3.3) and establishes, with Theorem A2,

the following

LEMMA A4. POP every f E C(k-l) [a,b] ® C(k-l) [a',b'] ,

thepe exists exaatZy one s E S2k,2k(~ x ~') top

whiah

356 C. de Boor

i = 1-k, ••• ,m+k; j = 1-k, ••• ,n+k •

We will denote this interpo1ating s by Rf •

Next, we define, in ana10gy to (6.1), (6.2), the

linear space

as the subspace of S ( .... x .... ') 2k,2k" " consisting of

those f for which

(3.5)

for (x,y) € 3R, j = O, ••• ,k-1 •

In the terms of the preceding section and of the

particu1ar Ai 's and ~j 's defined in (3.1) and (3.2)

* S2k,2k(7f X 7f') consists of those

for which

Appendix to Splines and Histograms

and

f = 0, for i = l-k, ••. ,O, m+l, ..• ,m+k Ai

f = 0, for j = l-k, ••. ,O, n+l, ••. ,n+k • llj

357

By the corollary to Theorem A2, we therefore conclude

* that S2k,2k(TI x TI') consists of those

fES 2k 2k (TI x TI') for which ,

Ai ~ lli f = 0, for all (i,j) i {l, ••• ,m} x {l, .•• ,n}.

On combining this statement with Lemma A4, we

obtain the following general version of the first

parts of Schoenberg's Theorems 3 and 4.

THOEREM A3. For a given reaZ m x n matrix

H = (hij ) ~ there exists exactZy one

f V(H) E S2k,2k(TI x TI') which satisfies (3.5)

and Yj xi

=! ! f(x,y)dxdy = hij , y. 1 x. 1 J- 1.-

i = 1, ... ,m; j l, ... , n

358 C. de Boor

This vo Zume matching sp Une V (H) can be computed

as

m n V(H) = I ly··cj>i 01j1 •

i=1 j=1 1J J

with

Here, A= C\cj>j) , B = (lliljlj) , with the cj>i 's

defined in (1.1), and the ljIi 's defined, correspon-

dingly, by

PROOF. It suffices to prove that

But this is clear since, certainly, the left hand

side contains the right hand side, while, on the

other hand, the dimension of the two spaces agree.

The existence and uniqueness of the interpolating

* V(H) implies, by Lemma A2, that dim S2k,2k(TI x TI')

* * ( , = mn, while mn = dim S2k(TI) x dim S2k TI )

REAL ANALYTIC FUNCTIONS AS RATlOS OF ABSOLUTELY

MONOTONIC FUNCTIONS

E.G. Straus

359

The problem we consider here is under what condi­

tions analytic functions which are positive on a segment

of the real axis can be expressed as ratios of two

absolutely monotonie functions, that is, functions all

of whose derivatives are non-negative on the given

segment.

The motivation for this question comes from the use

of generating functions of the form

where A is a given set of (non-negative) integers and

m is a set of (non-negative) multiplicities. It is a

clear that whatever functional equation we may get for

360 E.G. Straus

f A we can use the hoped for combinatorial interpretation

only if the Taylor coefficients turn out to be non-nega­

tive integers for all a E A. Let me illustrate this

with an example from a problem raised and solved by

Leo Maser:

PROBLEM: Divide the natural numbeps N = {0,1,2, ••• }

into two disjoint sets A3 B so that the sets of sums

of two distinct elements of A is the same as the set

of sums of two distinct elements of B (counting multi­

p Uai ties) .

The solution by Lambek and Maser [2] is brief and

elegant:

Set and assume

without loss of generality that 0 E A so that f A (0) = 1, 00

f (0) 0 N b h th i fA + fB -- \ xn -- (l_x)-l B = • ow y ypo es s l n=o

and

(1)

=

Real Analytic Functions as Ratios 361

If we write G(x) = fA(x) - fB(x) then (1) yields

or, by iteration

(3) 2 4 G(x) = (I-x) (l-x ) (l-x )

__ ~ q2(n) n L (-1) x

2n (l-x )

where q2(n) is the sum of the digits of n written to

the base 2. Thus we get the unique solution

1 1 fA(x) = 2 (G(x) + I-x) =

even

a x

so that A consists of those numbers with even sums of

digits to the base 2 ,and B of those with odd sums

of digits.

To see the intimate connection with the problem of

this talk, let us consider the Moser problem where A u B

is not necessarily the set of all natural numbers, A n B

is not necessarily empty and the elements may have multi­

plicity greater than 1. However for the sake of

362

analytic simplicity we assume m = O(eEa), a

E.G. Straus

Eb ~ = O(e )

for all a E A, bEB, E > O. If we set F = f A + f B

the equation (2) becomes

(2') G(x)F(x) = G(x2)

and (3) becomes (if we assume 0 E A, o '- B, m = 1) o

(3' ) 1 G(x) = ------=-----2n 2 F(x)F(x ) F(x ) ..•

which constitutes a representation of G(x) on the

interval [0,1) as the ratio of two absolutely monotonic

functions (the numerator being rather simple). Of course

the combinatorial problem is by no means settled by (3')

since m > 0, n ~ 0 will hold only provided a - 0

(4) n = 0,1,2, ....

We can sum this up as folIows.

THEOREM. The Lambek-Moser problem of dividing a given

set C of natural numbers with m = 1, m = O(eEC ) o c

for all c E C, E > 0 , into two sets A, B so that

the sums of two distinctly labelled elements are the

same sets with the same muUipUcities for A and B

Real Analytic Functions as Ratios 363

has at most one soZution given by

co n fA(x) = ~ (fc(x) + TTfc(x2 )-1)

n=o

co n f B (x) = ~ (fc(x) - 1T f c (x2 )-1)

n=o

co n If G(x) = TI f C(x2 ) -1 violates (4) then the Lambek-

n=o

Moser problem has no solution. In particular, if fC(x)

has a zero inside the unit disk then G(x) has poles in

the unit disk and the Lambek-Moser problem has no

solution. On the other hand, whenever

(5)

where Yi , ~i are positive natural numbers we get

and condition (4) is satisfied with A consisting of

the sums of even numbers of Y 's; B the sums of odd

numbers of y 's; and C of all finite sums of y 'so

364 E.G. Straus

There are also known examp1es [1] where some of the

exponents ~i in (5) are negative and (4) remains

satisfied. It wou1d be interesting to know whether all

solutions of the Lambek-Moser problem are of this form.

The answer is yes for finite sets [1] as we sha11 now

show.

My eoworkers (A. Fraenkel, B. Gordon, J. Se1fridge

[5], [1]) and I have examined the Lambek-Moser problem

for finite sets. Here equation (2') gives us a good

deal of information sinee F, Gare polynomials and

F(O) = 1. We see immediate1y that whenever G(S) = 0

then G(S2) = G(S)F(S) ~ O. Sinee S ~ 0 this is

possib1e on1y if S is a root of unity and henee both

G(x) and F(x) are eye1otomie polynomials. If we write

G(x) k Cli.e. ß.

= TT (1 - x ) / 1T (1 - x J) i=l j=l

we get

(6) F(x) = G(x2)/G(x) k Cl..e. ß.

= TI (1 + x ~) / TT (1 + x J) i=l j=l

as the ratio of two abso1ute1y monotonie funetions. One

entertaining eonsequenee of (6) is that whenever two

distinet sets A, B of order lAI = IBI = n have the

Real Analytie Funetions as Ratios 365

same sums of pairs of elements we get 2n = F(l) k-l 2 •

In other words the Lambek-Moser problem ean have

solutions in finite sets only if the number of elements,

n , is apower of 2. Equation (5) shows that all sueh

values of n are indeed possible. The finite Lambek­

Moser problem has thus been " redueed" to that of deter­

mining those eyelotomie polynomials (6) whieh themselves

are absolutely monotonie, that is have only non-negative

eoeffieients.

It is this problem whieh led T.S. Motzkin and me to

rediseover and reprove a theorem of G. Polya [4].

THEOREM (Polya). If P(x) = P(xl, ••• ,xn) is a poZyno­

miaZ such that p(x) _> 0 whenevep x. > 0, i = l, ... ,n , ~-

then thepe exist poZynomiaZs Q(x), R(x) with positive

coefficients so that P(x) = Q(x)/R(x) .

Fortunately Motzkin and I were not aware of this

theorem and as a result obtained more preeise quanti­

tative results. For polynomials in one variable with

P(x) > 0 for x > 0 it suffiees to eonsider the

irredueible real faetors whieh are either x + r, r > 0

or 2 2

x - (2r eos 8) + r with in whieh

eases the faetors are already absolutely monotonie; or

366 E.G. Straus

finally 2 2

Q(x) = x - (2r cos 8) + r ,

in which case we have

TI o < 8 < 2' r > 0

THEOREM (Motzkin-Straus [3]). Let n be the integer so

that (n-l)8 < TI < n8. Then a positive polynomial

divisible by Q(x) must be of degree d > n. If

n8 = TI then

xn + r n Q (x) = ----::--~--"'-----­

n-2 L (rn- 2- ksin(n_l_k)8)xk

k=o

1 sin 8

If n8 > TI then for each m3 0 < m < n we have

where

p (x) m

R (x) m

Q(x) p (x)

m R (x)

m

1 (sin n-2 + sin 28rxn- 3 + sin 8 8X ... n-2 sin n8 n-m (sin 8xm-2 + + sin(n-l)8r ) sin 8 sin m8 r

+ sin 28rxm- 3 + m-2 ... + sin(m-l)8r )

Real Analytic Functions as Ratios 367

Moreover if cos a ~ m/ (m+2) then m Q(x)(x+r) has

positive coefficients so that every polynomial which is

positive on the positive axis can even be expressed as

a positive polynomial divided by a totally positive

polynomial, that is a polynomial whose zeros are

positive.

This raises a number of questions

1. For every polynomial F(x)

e:cists a minimal p01.Vep nF of

of the foPm (6) there

(X+l) so that nF

F(x) (X+l) has non-negative aoeffiaients. If deg F = n what aan we say about nF ?

We can answer this in part.

LEMMA: Let ~2m (x) be the irreduaible ayalotomia

polynomial of order 2m (degree ~(m») then

2 n~ ~ 3m /4 •

2m

Thus for F = ~2 ~2 ••• ~2 we get ml m2 m,e.

3 2 222 nF ~ '4 (mI + .•• + m,e.) ~ cn (log log n) • On the other

hand if F(x) = x2n _ xn + 1 = x3n - 1 xn + 1

then

2 nF > log 2 n(n-l). Thus the upper bound for nF seems

368

to be of the order of magnitude of

faetor of magnitude 2 (log log n) •

2 n

E.G. Straus

up to a possible

It would be possible

to eonvert these estimates into rough estimates of the

number of pairs of sets A, B of natural numbers with

max e < n whieh have the same sums two at a time. eEAuB

11. To what extent ean the resuZts of PoZya-Motzkin­

Straus be extended to generaZ power series?

The results do not extend to all funetions, f,

analytie on an interval [a,b] of the real axis with

f(x) > 0 for a < x < b. This follows from a result of

S. Bernstein to the effeet that an absolutely monotonie

funetion whieh is analytie on [a,b] is in faet analytie

in the disk lx-al< b - a. Thus, if we eould write

f = g/h where g and h are absolutely monotonie then

f must be meromorphie in the disk lx-al< b - a. Sinee

a funetion, f, meromorphie in, say, lxi < 1 with

f(x) > 0 for 0 < x < 1 ean be expressed as the ratio

of two funetions g(x)/h(x) holomorphie in the unit disk

with g(x) > 0, h(x) > 0 for 0 < x < 1 it suffiees to

eonsider the following question.

111. Let f(x) be hoZomorphie in lxi < l. Can we

express f as the ratio g/h of two absoZuteZy mono­

tonie funetions g, h whieh are hoZomorphie in

Real Ana1ytic Functions as Ratios

lxi< I? Can we insist that, in addition, h have

onZy negative zeros in lxi< 1 ?

The answer is yes to both questions if f has

a finite number of zeros in lxi < 1 and hence yes

we look for the representation g/h on1y in a disk

lxi < r < 1 •

369

on1y

if

In its comp1ete genera1ity this question as weIl as

its genera1izations to ana1ytic functions of severa1

variables remain unanswered.

REFERENCES

1. Fraenkel, A.S. and Gordon, B. and Straus, E.G.: On the determination of sets by sets of sums of a certain order, Pacific J. Math. 12 (1962), 187-196.

2. Lambek, J. and Moser, L.: On some two way c1assifications of integers, Can. Math. Bu11. 2 (1959), 85-89.

3. Motzkin, T.S. and Straus, E.G.: Divisors of polynomials and power series with positive coeffi­cients, Pacific J. Math. 29 (1969), 641-652.

4. Po1ya, G.: Über positive Darstellung von Polynomen, Vierte1jahvsschrift Zürich 73 (1928), 141-145.

370 E.G. Straus

5. Se1fridge, J.L. and Straus, E.G.: On the deter­mination of numbers by their sums of a fixed order, Pacific J. Math. 8 (1958), 847-856.

INVERSE THEOREMS FOR APPROXIMATION BY POSITIVE

LINEAR OPERATORS

Ron DeVore

371

We are interested in studying the relation

between the smoothness of a function and its degree of

approximation by means of a sequence (L ) n

of positive

linear operators defined on aspace of continuous

* functions C[a,b] or C [-n,n]. Our main interest

is in what inferences can be made about the smoothness

of a function f when we assume something about the

rate of decrease of 11 f - L (f)11 . Such a result is n

customarily called an inverse theorem of approximation

while a result which estimates 11 f - L (f)11 in terms n

of the smoothness of f is called a direct theorem.

Direct theorems are relatively easy to obtain and

are known for the classical examples. On the other

hand, inverse theorems are much more difficult to prove

372 R. DeVore

and indeed may not even hold. The eustomary way of

proving inverse theorems is to use the ideas used by

s. Bernstein in his proof of the inverse theorems for

approximation by trigonometrie polynomials.

Bernstein's teehnique relies on knowing estimates

for suitable derivatives of L (f) (the analogue cr n

Bernstein's inequality). This preeludes the handling

of general sequenees (L ) n

sinee L (f) need not even n

be differentiable in the general ease.

In this work, we replaee the eonditions on the

derivatives of L (f) by suitable eonditions on the n

eoneentration of "mass" of L n

For an example,

suppose (L ) n

is a sequenee of positive eonvolution

operators, i. e.

L (f,x) n

I 1T = - f f(X+t)d~ (t)

1T n -1T

with d~ a non-negative, even Borel measure on n

with unit masse Let

<1>2 1T

t2d~ (t) = f n n -1T

[-1T ,1T]

Inverse Theorems 373

then we can show

THEOREM. Let 0 < a < 2. If there is an M > 0 with

(1)

and

(2)

~n < M < +00, n = 1,2, •..

~n+1 -

TI

f -TI

then * f E Lip a if and only if "f - L (f)" = O(~a) • n n

The assumption (2) is the restrietion on the con­

centration of mass. The assumption (1) is a1ways

needed for general inverse theorems and it essentia11y

guarantees that the sequence (L ) n

is not to sparse,

i.e. there are sufficient1y many L n

As an examp1e of this theorem, let t '" 0 and n

Each L can be written as convo1ution with the n

measure d~ which is pure1y atomic with masses n

at each of the points -t and t n n

The theorem

TI

2

374

shows that if

(3)

then

(4)

imp1ies

(5)

t _n_ < M < -t<x> t n+1 -

1I1I 2 (f , x) 11 t n

n = 1,2, •••

O(ta ) n

where 2 lI t (f,x) = f(x+t) + f(x-t) - 2f(x) •

R. DeVore

It can

also be shown that (3) is a necessary condition for (4)

to imp1y (5) in the sense that if (3) does not hold

then there is a function f which s~tisfies (4) but

not (5).

We can also use our technique to prove inverse

theorems for operators that are not given by con-

vo1ution. In this case,

and (2) is rep1aced by

~2 is replaced by n

Inverse Theorems

4 L «t-x) ,x) n

0(4) \x» n

This gives, for example, the inverse theorems for

Bernstein polynomials which were given by H. Berens

and G.G. Lorentz. Namely, a necessary and sufficient

* condition for f to be in Lip a is that

If(x) - L (f,x) I < M {x(1-x)}a/2 n n

for some constant M > 0 •

375

377

LACUNARY INTERPOLATION BY SPLINES

A. Meir & A. Sharma

1. In 1955, J. Sur~nyi and P. Turan commenced the

study of what they called (0,2) interpolation. By

(0,2) interpolation we mean the problem of finding the

algebraic polynomial of degree ~ 2n-l , if it exists,

whose values and second derivatives are prescribed on

n given nodes.

From the above article of Suranyi and Turan and

from the subsequent articles of Balazs and Turan it is

transparent that the problem of explicit construction

of the (0,2) interpolatory polynomials is a difficult

one. Moreover, their methods apply only to special sets

of nodes, which do not include the equidistant case.

More recently Schoenberg has initiated the study

of the so-called g-splines in connection with the pro­

blem of lacunary interpolation by splines. He showed

378 A. Meir & A. Sharma

that under certain conditions the interpo1atory g­

sp1ines exist and are unique.

In this paper we obtain error bounds for some

c1asses of quintic sp1ines which interpo1ate to (0,2)

data on equidistant knots. Natura11y, such quintic

sp1ines are deficient sp1ines. According to the demands

of continuity and the end conditions required, we obtain

different c1asses of quintic sp1ines. Our method of

proof leads to an a1gorithm for the numerica1 evaluation

of the interpo1atory sp1ines on the basis of the given

data.

2. DEFICIENT QUINTIC SPLINE INTERPOLATION.

For n = 2,3,4 ... we sha11 denote by the

c1ass of quintic sp1ines S(x) on [0,1] having the

fo11owing two properties:

(i) 3 S(x) E C [0,1]

(ii) S(x) is a quintic in [~v+1] ° 1 1 n' n ' v = " ••• ,n- .

It is c1ear that S(35) is a 2n + 4 n,

linear subspace of C3 [0,1] and if

dimensional

S(x) E S(35) , then n,

Lacunary Interpolation by Sp1ines 379

S(x) n-1 v 4 v S

= q(x) + L {c (x - -) + d (x - -) } v=l v n + v n +

where q(x) is a quintic and cv ' dv are constants.

THEOREM 1. Por every odd integer n and for every

given set of 2n + 4 reaZ numbers

{f f f 'f" f" f"·f'" f"'} '1'···" '1'···" , o non 0 n

there exists a unique S(x) € S(3) such that n,S

(1) v

S(-) = f n

v S" (-) = n

(2)

(3) S'" (0) =

f" v

f'" o

v

v

= O,l, ... ,n

= O,l, .•• ,n

S"'(l) = f'" n

THEOREM 2. Let f E C4 [0,11 and n

Then for the unique quintic spZine

an odd integer.

S (x) satisfying n

v (1), (2) and (3) with f v = f(n)'

v f" = f" (-) v n '

v = 0,1, ... ,n; f'" = f'" (0), o

f'" = f'" (1) , we have n

(4)lls(r)-f(r)11 < 7Snr - 3w (!.) + 8nr-41If(4)~ r = 0,1,2,3 n 00 - 4 n 00'

380

where w (.) 4

A. Meir & A. Sharma

denotes the moduZus oi continuity oi f(4).

THEOREM 3. Let f E C4rO,I] and n any positive

integer., h = n-l Then there exists a unique quintic

spZine S*(x) E S(3) n n,S such that

(1' )

(2')

(3' )

Moreover.,

* S (vh) = f(vh) , n

*" S (vh) = f"(vh) , v = O,l, .•• ,n n

*' S (0) = f' (0) , n

*'" S (0) = f"'(O) • n

r = 0,1,2,3 •

Remark: In general there is no unique quintic periodic

spline S (x) E S(3) which satisfies (1) and (2), even n n,S

if the data {f} and {f"} are periodic. v v

This paper will appear in full in the S.I.A.M.

Journal of Numerical Analysis.

STABILITY PROPERTIES OF TRIGONOMETRIe INTERPOLATION

OPERATORS

P.D. Morris & E.W. Cheney

381

Consider the spaee C of all 2n-periodie

eontinuous real funetions, and the subspaee TI of all

n-th order trigonometrie polynomials. The index n is

held fixed, and the spaees are endowed with the usual

supremum norm. Any operator L : C + TI whieh ean be

m written in the form Lx = L x(sk)Yk with ° 2 sk < 2n

1

and Yk E TI is said to be carried by the point set

If Lx = x for all x E TI , then L

is a projection of C onto TI.

defined to be the set of points

for k = O, ••• ,2n .

The uniform grid is

-1 t k = kn(2n + 1)

THEOREM 1. Let S be a set of 2n + 2 points con-

382 P.D. Morris & E.W. Cheney

taining the uniform grid. Among all the projections

from C onto TI carried by s, the interpolating

projection carried by the uniform grid has least norm.

THEOREM 2. Let S be a set of points containing the

uniform grid and containing at least one point between

each two points of the uniform grad. Then the inter­

polating projection on the uniform grid is not minimal

in norm among the projections carried by s.

THEOREM 3. There exists a set S of 2n + 4 points

containing the uniform grid with the property that the

interpolating projection carried by the uniform grid

is not minimal among the projections carried by s.

CHEBYSHEV SEMI-DISCRETE APPROXIMATIONS FOR LINEAR

PARABOLIC PROBLEMS

Richard S. Varga

Consider the approximate solution of the linear

system of ordinary differential equations

J d!!.( t) A!!.(t) + r Vt > 0 , = -dt (1)

1 !!.(O) ~

= u

383

where A is a given n x n Hermitian and positive

definite matrix, and where rand u are given

n-vectors. Such linear systems arise, for examp1e, in

the so-ca11ed semi-discrete numerica1 approximations to

linear heat-conduction problems in which the spatia1

variables are suitab1y differenced, but the time

variable is 1eft continuous. The sQ1ution of (1) is

c1ear1y given by

384 R.S. Varga

(2) ~(t) -1 1

= A ~ + exp(-tA){~ - A- r} Vt ~ 0

In contrast with the usual Pade methods, based on

Pade rational approximations of -x e in the neighbor-

hood of x = 0 , we consider here approximations of

w(t) of (2) based on Chebyshev rational approximations

of -x e on [0,+00) , defined as folIows. If 1f

m

denotes all real polynomials of degree at most m

and 1f analogously denotes all real rational m,n

functions r (x) = p(x)/q(x) m,n

define

(3) A m,n

with q E 1f n

r E 1f } m,n m,n

for all nonnegative integers with 0 < m < n , and let

f = ß /4 E 1f be such that m,n m,n m,n m,n

(4) A m,n

Then, the Chebyshev semi-discrete approximation,

w (t), of the solution ~(t) of (1) is defined by -nl,n

Vt > 0 .

Chebyshev Semi-Discrete Approximation

Using i. e. ,

shown that

n I Iv1 12 , it 1s then

i=1

and, as (6) is valid for all t ~ 0 , the Chebyshev

sem1-d1screte approximation w (t) 1Il,n

can be regarded

385

as a one-step approximation of ~(t) for any t > 0 .

The accuracy of the Chebyshev semi-discrete method

depends, from (6), on A , and it is obvious from m,n

(3) that

(7) o < A < A < n,n - n-l,n-< A

o,n

Moreover, it is known that the A 's have goemetric o,n

convergence to zero, i.e.,

(8) 11m (A )l/n = 1 o,n 3 n--)<lO

Finally, it is natural to ask which entire func­

tions fex) (having nonnegative coefficients in its

Taylor's expansion about x = 0) possess the property

that, if

386 R.S. Varga

(9) A - inf {ll f (l) - r (x)II L [0 co] : r € 7f } m,n x m,n co' m,n m,n

then there exists a q > 1 such that

(10) 1im (A )l/n < 1 < 1 • o,n - q

n~

Sufficient conditions are given for this, and a re­

stricted converse is also estab1ished.