NECESSARY CONDITIONS FOR THE CONVERGENCE OF ...

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 255, November 1979

NECESSARY CONDITIONS FOR THE CONVERGENCE OF CARDINAL HERMITE SPLINES AS THEIR DEGREE TENDS TO INFINITY BY

T. N. T. GOODMAN Abstract. Let §„_, denote the class of cardinal Hermite splines of degree n having knots of multiplicity í at the integers. In this paper we show that if /„ -»/uniformly on R, where/, e S^ i„ -» oo as n -» oo, and/is bounded, then/is the restriction to R of an entire function of exponential type < s. In proving this result, we need to derive some extremal properties of certain splines &„j e Sn>J, in particular that ||©„^||00 minimises \\S\\X over S e

Sn>,with ||S«->|U= llSgn..

1. Introduction. For n = 1,2, . . . and 1 < j < n, let

9„ = {/ G C"-*(R): f\(v, v + I) e C'Yv,

v + 1)] and

/(n_1) absolutely continuous on (v, v + 1), V v G Z}. We let Sn^ denote the set of all cardinal spline functions of degree n in

§„,, = {5 G C-'(R):

S|(r, r + 1) £ wmV r e Z},

where 7rndenotes the set of all polynomials of degree at most n. Throughout this paper, ||/|| will denote ess supxeR|y(jc)|. In [6] Lipow and Schoenberg have shown that for odd n, 1 < s < \{n + 1), and any function / with f(v) of power growth on R, v = 0,1, . . ., s — 1, there is a unique SnySG S„^ of power growth such that S^ interpolates /(,,) at the integers. In [8] Marsden and Riemenschneider have shown that if / is the Fourier-Stieltjes transform of a measure on (sw, jw), then S^J-*f-r) uniformly on R as n —>oo, v = 0,1, . . . , s — 1. The case s = 1 had previously been proved by Schoenberg [10] who established in [11] the partial converse that if/is bounded on R and Snl -»/uniformly on R as n —*oo, then/is the restriction to R of an entire function of exponential type < it. In §4 of this paper we generalise Schoenberg's result by showing, in particular, that for any s = 1,2,. . . , if / is bounded on R and S„¿ ->/ uniformly on R as n -^ oo, then /is the restriction to R of an entire function Received by the editors August 18, 1978. AMS (MOS) subject classifications (1970). Primary 41A15; Secondary 41A05. Key words and phrases. Cardinal spline interpolation,

cardinal Hermite splines, Euler splines,

Chebyshev polynomials. © 1979 American Mathematical 0002-9947/79/0000-0509/$03.75

231

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Society

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T. N. T. GOODMAN

of exponential type < sir. To establish this result we need some extremal properties of certain splines êBjJ G SB>Jwhich may be regarded as generalisations of the Euler splines employed in [11]. For odd s these were defined by Cavaretta in [1]. In §2 we define &n¡s for even * and show that for all s,

f G §^, 11/11 < 1 = \\S„J and \\f\ \fw(p +)| < \&ikJ(v +)|, In [1] Cavaretta S G Sn>Jwith

< ||gW||implies

V v ||(* + x) =

V ; -1)

s odd, ,

s even,


CONVERGENCE OF CARDINAL HERMITE SPLINES

237

where

«-Í1,

Ífai>0'

(3-9)

*

if a, = 0.

V '

\ 0,

We are now in a position to prove our result.

Theorem 2. If S G Sn>ísa/w/ieï ||,S|| < 1, then \\Sin)\\ < \\S%>\\. Proof. Take ßx, . . ., ßs as in (2.2). By (2.3) we know the nonzero /?,-, j = 1, . . ., s, are symmetric about ~ and so IL//?,, . . ., ßs; X) is a reciprocal polynomial in A. If n and 5 are both even or both odd, then /?, = 0. Otherwise ßx > 0. Thus in all cases, n„(/?„ . . ., ßs; X) is a. polynomial in X of even degree and so

ii,(A,.,.,A;(-i)')*aSince (3.4) is satisfied, we may define the 'fundamental spline' Lr for r = 1, . . . , s. Then for any S G SMÍ satisfying \\S\\ < 1, we have from (3.6), |S(n)(*)|

=

2

2 s(k + ß,)LP(x - k)

r= 1 k = -oo

< 2

2

|Lin)(* - *)|>

VxGR

(3.10)

r= 1 /c = -oo

But it follows from (3.9) and (2.2) that equality is attained in (3.10) for

S-&„-

D

For j = 1 this result was proved by Schoenberg [11], and for s = n the result follows immediately from the properties of Chebyshev polynomials. It is clear from the proof of Theorem 2 that the condition ||5|| < 1 in the statement of the theorem can be replaced by the weaker condition

\S(k + ßi)\ < 1,

V k G Z, i - 1,....

s.

4. Limits of cardinal splines. We need a further property of ET-splines. Lemma 2. For s = 1,2, ...

, there are constants Ks such that ||S^||

s and v = 0, . . ., n.

Proof. First suppose j is odd, s = 2/ — 1. It follows from the work of [1] that for any n > s, &n,s = $«,, + MlS„-2,l + • • * +M,-lS„_2,

+ 2>1,

(4.1)

where ju,, . . ., /*,_! are chosen to minimise ||Sn^||. We first consider odd n > s. Then it follows from (4.1) and (2.6) that we

may write

Ks = H)("+,)/V|W|,


238

T. N. T. GOODMAN

where y

9nV >

A

cos(2/- - 1)ttx

(2r-l)"+1

' ,-.

(n) £ +

(n) ™ cos(2r - l)nx

cos(2r - 1)ttx

+X'-r?.(2r-l)—3

- 2 ^'"J^T'O r-l

(Ir-l)-1

+ M->(2r - l)2+ • • • +X,I(2r - l)2'"2},

(2r — 1)

and X\"\ . . ., Xf"?lare chosen to minimise ||„||. Let A], . . . , X,_ ] be the unique solution of the equations

1 + (2r - 1)2A, + • • • + (2r - 1)2'"2A,_, = 0,

r = 1, ...,/-

1.

Let

*. 0 as n -* oo. Since ||6,|| < ||«,||, ||(2/ - 3)"+1 a = 1 + (2/ - 1)2a, + • • • + (2/ - I)2'-2*,-!

Now for each n, there is an integer/,

•H2/-

=*=0

as n

oo.

1 < / < It — 1, such that

i )ancos> > °»

and so

P'-')"'M^)|>w+0([^t]") So 3 5 > 0 such that ín+1||„||>ó\

VOí.

(4.2)

Writing , ,

£

gn{) ' rh

cos(2r - \)ttx

(2r _ !)-

'


CONVERGENCE OF CARDINAL HERMITE SPLINES

we have

||¿>)||(l+± for« = 1,2, ...

+ ±+...) s and v < n. Thus Ks

IS&II= ll^ll/IWI < -y («)",

Vn > s, v»/wi.


240

T. N. T. GOODMAN

where

*„(*)- 2 ^^sin2^(x-i) k=l

+ A1 s and k < n - s.

Proof. Take S in Sn¡swith ||S|| < 1. Then by Theorem 2, ||S(">|| < ||S(„nJ||. So by Theorem 1, |S(*)(y+)|