The Sampling Theorem and Several Equivalent ... - Springer Link

Journal of Computational Analysis and Applications, Vol. 2, No. 4, 2000

The Sampling Theorem and Several Equivalent Results in Analysis J. R. Higgins,1 G. Schmeisser,2 and J. J. Voss2 The authors take pleasure in dedicating this article to Professor P. L. Butzer

First we show that several fundamental results on functions from the Bernstein spaces B p (such as Bernstein's inequality and the reproducing formula) can be deduced from a weak form of the classical sampling theorem. In § 3 we discuss the mutual equivalence of the sampling theorem, the derivative sampling theorem and a harmonic function sampling theorem. In § § 4±6 we discuss connections between the sampling theorem and various important results in complex analysis and Fourier analysis. Our considerations include Cauchy's integral formula, Poisson's summation formula, a Gaussian integral, certain properties of weighted Hermite polynomials, Plancherel's theorem, the maximum modulus principle, and the PhragmeÂn±LindeloÈf principle. KEY WORDS: Sampling theorem; Cauchy's integral formula; Poisson's summation formula; Fourier analysis; complex analysis.

1.

INTRODUCTION

In recent years several authors have shown that there is a strong relationship between the well known classical sampling theorem (often called the Whittaker±Kotel'nikov±Shannon sampling theorem) and other fundamental theorems of real and complex analysis such as Poisson's summation formula and Cauchy's integral formula (see, e.g., [6; 8; 15] and the surveys [5; 7; 12, Ch. 9]). Our present purpose is to discuss relationships between the sampling theorem and several other fundamental results of analysis. We start by deriving some consequences of the sampling theorem, and then go on to investigate groupings of equivalent results. 1 2

Division of Mathematics and Statistics, Anglia Polytechnic University, Cambridge, England. Mathematisches Institut, UniversitaÈt Erlangen-NuÈrnberg, D-91054 Erlangen, Germany. 333 1521-1398/00/1000-0333$18.00/0 ß 2000 Plenum Publishing Corporation

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Higgins, Schmeisser, and Voss

The material is organized around the sampling theorem since this seems to be natural. The theorem does play a major roÃle in § 2, and, of the subsequent equivalence groupings, it alone is common to all. Perhaps this just re¯ects a convenience of presentation, but it seems undeniable that the theorem makes its presence felt in many areas of analysis. Two results A and B will be considered ``equivalent'', a relationship denoted as usual by A () B, when each can be obtained from the other by straightforward methods. Such methods must, of course, avoid circular reasoning. Two equivalent results are often considered to have ``comparable depth'' (see [11, p. 563] for an interesting account of this concept as it arose in number theory), and it is this comparability of depth that accounts for much of the interest in studying equivalence groupings. In proving the implications which form the substance of this study, certain principles of analysis will inevitably be involved. They will be referred to as ``side results'', and we have tried to ensure that their proofs do not introduce circularity. For example, it will be noticed that the hypotheses of the results discussed in §§ 3, 4 involve the space denoted by Bp , whereas b p (see the De®nition below); if the classical Paley± those in §§ 5, 6 involve B  Wiener theorem [1, p. 103] were an allowable side result, we could deduce that these two function spaces are identical. However, proofs of the Paley± Wiener theorem depend ultimately on the Cauchy integral theory and would therefore introduce circularity. It may be mentioned that if an essentially ``real variable'' proof of the Paley±Wiener theorem were available, it would become a viable side result, and would undoubtedly lead to a more ecient organization of the equivalence groupings presented here. In § 2 we start by assuming that the sampling theorem holds for the  space B 2 (see § 2 for the de®nition), which is a rather amenable space to work with. We then extend to the more general spaces Bp , and some fundamental results follow; for example, Bernstein's inequality, an ` p estimate for sums of sampled values, Tschakalo€ 's interpolation formula, a translation invariance principle for B p and a ``reproducing'' formula. The extension procedure and many of the consequent results are needed in the sequel. In § 3 we discuss the mutual equivalence of the sampling theorem, the derivative sampling theorem and a harmonic function sampling theorem. In § 4 we discuss the mutual equivalence of the sampling theorem and the Cauchy and Poisson formulae. This grouping has been studied by Butzer, Hauss, Ries, and Stens, particularly in [4] and [6]. In § 4 we ®nd it necessary to modify some of their conclusions, particularly in connection with Poisson's formula. In § 5 we discuss the mutual equivalence of the sampling theorem, Cauchy's integral formula, a Gaussian integral, certain properties of the

The Sampling Theorem and Several Equivalent Results in Analysis

335

weighted Hermite polynomials, and Plancherel's theorem. Here, several proofs are to be found in the book of Titchmarsh [20, Ch. 3], and when this occurs we shall give only brief explanatory details. In § 6 we discuss the mutual equivalence of the sampling theorem, the PhragmeÂn±LindeloÈf principle and the maximum principle. 2.

THE CLASSICAL SAMPLING THEOREM AND SOME OF ITS CONSEQUENCES

We shall make use of the function spaces speci®ed in the following de®nitions. De®nition 1. For  > 0 and p 2 ‰1; 1Š; let B p be the class of all entire functions f whose restriction to R belongs to Lp …R†, that is, … 1 p if 1  p < 1; kf kp :ˆ jf…x†jp dx < 1 ÿ1

kfk1 :ˆ supjf…x†j < 1 x2R

if p ˆ 1;

and which satisfy an inequality

ÿ
jf…z†j  M exp j= zj

for z 2 C

…1†

with a number M depending only on f: This de®nition readily implies that and

B p  Bp

for   

B p  Bq

for 1  p  q  1:

We shall also use the following subspaces of B 2 : b 2 comprises all functions f : C ! C whose De®nition 2. The class B  2 restriction to R belongs to L …R†, and are such that … f…z† ˆ eizt …t† dt …2† ÿ

for some  2 L2 …ÿ; †.  The class B 2 comprises ÿ1 f…x† ˆ O…jxj † as x ! 1:

all functions f 2 B 2

which satisfy

336

Higgins, Schmeisser, and Voss

b 2 is continuous at all points z 2 C. Clearly, every function f 2 B  Moreover, a straightforward estimate of the integral in (2) shows that f is analytic in C and satis®es (1); this is the ``trivial'' half of the classical Paley± Wiener theorem. Hence, b2  B2 : B  

…3†

b 2 and B 2 are identical; this is the ``non-trivial'' half of the Paley± In fact, B   Wiener theorem. However, in the circumstances described here we cannot invoke the Paley±Wiener theorem indiscriminately. This is because its proof (see e.g., [18]) depends ultimately on the Cauchy integral theory, and we must be careful to avoid circular reasoning. Likewise we always seek realvariable proofs of the necessary side results.  The class B 2 allows us to relax the hypothesis of certain statements (see Theorem 2 below). The most fundamental result of this paper is the sampling theorem. In terms of the spaces B p and the sinc function 8 > < sin…z† for z 2 C n f0g z sinc z :ˆ > : 1 for z ˆ 0 it may be stated as follows. p …1  p < Theorem 1. (The Classical Sampling Formula). Let f 2 Bw 1†: Then X n sinc…wz ÿ n†; …4† f f…z† ˆ w n2Z

the convergence being absolute and uniform on compact subsets of C. Let X be a subspace of B p . We shall use the notation CSF…X † to mean that we consider the Classical Sampling Formula (4) with w ˆ = for functions f 2 X . First we show that the sampling formula gives access to various remarkable properties of the spaces B p . It turns out that even the  weaker form CSF(B 2 ) will suce. Theorem 2. Let f 2 B p with 1  p  1: The following statements are  consequences of CSF(B 2 ) (i) The generalized Bernstein inequality: kf 0 kp  kfkp :

The Sampling Theorem and Several Equivalent Results in Analysis

337

Moreover, B p is invariant under di€erentiation. (ii) An ` p estimate: If h > 0 and p 6ˆ 1; then h

X n2Z

!1=p jf…nh†jp

 …1 ‡ h†kfkp :

(iii) Tschakalo€ 's interpolation formula [12, p. 60]: 0 1 n 0 X f …0† f…0† z A; ‡ ‡ …ÿ1†n f f…z† ˆ sin…z†@  z  n…z ÿ n† n2Znf0g

…5†

…6†

where the right-hand side is de®ned for z ˆ n= …n 2 Z† by continuation. The convergence is absolute and uniform on compact subsets of C. (iv) Invariance under translation: f…
‡ c† 2 B p for any c 2 C: Moreover, there exists a positive function K, not depending on f and p, such that3 kf…
‡ c†kp  K…=c† kfkp : (v) The generalized reproducing formula: If p 6ˆ 1, then   …  1 …z ÿ t† f…t† dt …z 2 C†: sinc f…z† ˆ  ÿ1 

…7†

…8†

Proof. (i) First we suppose that in addition f…x† ˆ O…jxjÿ1 † as x ! 2 1: Then CSF(B  ) applies. Moreover, we may di€erentiate both sides of (4) and interchange summation and di€erentiation on the right-hand side. At z ˆ =…2†, we obtain    4 X …ÿ1†n‡1 n f ˆ 2 : f0 2  n2Z …2n ÿ 1†2 

…9†

But this formula holds for an arbitrary f 2 B p as well. In fact, for " 2 …0; 1=2†, the function ÿ
f" : z 7ÿ!f …1 ÿ "†z sinc…"z=† 

belongs to B 2 . Hence (9) applies to f" : Letting " ! 0; we ®nd that (9) holds for f itself. 3

The function K…x† :ˆ exp…jxj† is best possible (see [1, Thm. 6.7.1]).

338

Higgins, Schmeisser, and Voss

Next, applying (9) to the function f…
‡ x ÿ =…2††; where x 2 R; we obtain   4 X …ÿ1†n‡1 …2n ÿ 1† 0 ; f x‡ f …x† ˆ 2  n2Z …2n ÿ 1†2 2 and so

0   X f …x† 1  4 f x ‡ …2n ÿ 1† :  2 2 2 n2Z …2n ÿ 1†

…10†

Besides, applying (9) to cos…x†, we ®nd that 4 X 1 ˆ 1: 2  n2Z …2n ÿ 1†2

…11†

Hence the right-hand side of (10) is a convex combination of the moduli of certain values of f: Now Bernstein's inequality for p ˆ 1 follows immediately. For 1  p < 1; the convexity of jxj 7! jxjp implies that 0 p   p X f …x† 1  4 f x ‡ …2n ÿ 1† : …12†  2 2 2 n2Z …2n ÿ 1† Since

 p …1 …1  f x ‡ …2n ÿ 1† dx ˆ jf…x†jp dx ˆ kfkpp ; 2 ÿ1 ÿ1

we readily deduce from (11) and (12) that kf 0 =kp  kfkp : The derivation of (10) allows to replace x by z 2 C: But then the inequality jf…z†j  M exp…j= zj† implies that jf 0 …z†j  M exp…j= zj†: Hence f 0 2 B p : (ii) A proof of (5) using the generalized Bernstein inequality, the HoÈlder inequality and the inequality j kxk ÿ kyk j  kx ÿ yk as the only nontrivial tools is given in [14, p. 123]. (iii) Tschakalo€ 's interpolation formula is obtained by applying the classical sampling formula to …f…z† ÿ f…0††=z which represents a function  in B 2 ; see [1, p. 220]. (iv) It is enough to prove (7) for c ˆ iy with y 2 R: By substituting z ˆ =…2† ‡ iy and introducing 1 1 =2 ‡ it …n 6ˆ 0†; ; An …t† :ˆ A0 …t† :ˆ n …2n ÿ 1†=2 ‡ it j=2 ‡ itj

The Sampling Theorem and Several Equivalent Results in Analysis

339

we deduce from (6) that 0

jf…=…2† ‡ iy†j  cosh…y† jf …0†=j ‡

X

! An …y†jf…n=†j :

n2Z

Applying this inequality to f…
‡ x ÿ =…2†† and de®ning X An …t†; A…t† :ˆ 1 ‡ n2Z

which is easily seen to exist, we obtain  X An …y†  1 1 0   …2n ÿ 1† jf…x ‡ iy†j  f xÿ ‡ f x‡ : A…y† cosh…y† A…y†  2 A…y† 2 n2Z The right-hand side is again a convex combination. For calculating the norm with respect to x of the left-hand side, we may argue as in the proof of the generalized Bernstein inequality and use the latter for estimating the ®rst term on the right-hand side. This leads to kf…
‡ iy†kp  A…y† cosh…y†kfkp ; which has the desired form.  (v) Let f 2 B 2 : Applying (4) with w ˆ = to f…
‡ t†; where t is a real parameter, and replacing < z by x ÿ t and = z by y afterwards, we obtain    X n …x ÿ t ‡ iy† ‡ t sinc ÿn : f f…x ‡ iy† ˆ   n2Z Next we integrate both sides over ‰0; =Š with respect to t and note that integration and summation may be interchanged since f…x† ˆ O…jxjÿ1 †. Therefore    X … = n  …x ÿ t ‡ iy† f…x ‡ iy† ˆ ‡ t sinc ÿ n dt f    n2Z 0 ˆ

X … …n‡1†= n2Z

n=



 …x ‡ iy ÿ t† f…t† sinc dt 

  …x ‡ iy ÿ t† sinc f…t† dt : ˆ  ÿ1 …1



This shows that (8) holds for f 2 B 2 :

340

Higgins, Schmeisser, and Voss

Now let f be an arbitrary function from B p and let " 2 …0; 1†: Then ÿ
f" : z 7ÿ! f …1 ÿ "†z sinc…"z=† 

belongs to B 2 , and so, by what we have proved already,   …  1 …z ÿ t† f" …t† dt sinc f" …z† ˆ  ÿ1      … 1  1 …z ÿ t=…1 ÿ "†† "t sinc sinc f…t† dt: ˆ …1 ÿ "†  ÿ1  …1 ÿ "†

…13† …14†

For ®xed z, the modulus of the ®rst sinc expression in (14) has an upper bound of the form A B ‡ jz ÿ tj with positive numbers A and B that do not depend on t and "; the modulus of the second sinc expression is bounded by 1. Now, when in (13) we pass to the limit " ! 0, we obtain f…z† on the left-hand side, while Lebesgue's theorem of dominated convergence applies to (14) and yields the right-hand side of (8). This completes the proof. & In the proofs of statements (i) and (v) we have extended certain results from B 2 to B p via an auxiliary function f" : The following two lemmas allow further applications of that kind. ÿ
Lemma 1. Let f 2 B 1 and f" …z† :ˆ f …1 ÿ 2"†z sinc2 …"z† with " 2 ‰0; 1=4†: Then …1 …1 f" …x† dx ˆ f…x† dx …15† lim "!0

ÿ1

and for any h > 0; lim

"!0

X n2Z

ÿ1

f" …nh† ˆ

X

f…nh†:

…16†

n2Z

Proof. Clearly, as " approaches zero, f" …z† converges pointwise to f…z† and so does ÿ
ÿ
f" z=…1 ÿ 2"† ˆ f…z† sinc2 "z=…1 ÿ 2"† : Hence, substituting t :ˆ …1 ÿ 2"†x on the left-hand side of (15), we obtain

The Sampling Theorem and Several Equivalent Results in Analysis

341

the ®rst statement as a consequence of Lebesgue's theorem of dominated convergence. For a proof of (16), we introduce the intervals Im :ˆ ‰mh; …m ‡ 1†h†; where m 2 N 0 , and write …1 ÿ 2"†nh ˆ: tn for short. Then f" …nh† ˆ f…tn † sinc2 …"nh†: Moreover, for every tn 2 Im ; we have … …m‡1†h jf…tn †j  jf…mh†j ‡ jf…tn † ÿ f…mh†j  jf…mh†j ‡ jf 0 …t†j dt : …17† mh

Hence 1 X

jf" …nh†j 

nˆ0

1 X

" jf…mh†j ‡

… …m‡1†h mh

mˆ0

! 0

jf …t†j dt

X




# 2

sinc …"nh† :

tn 2Im

…18† Obviously, tn 2 Im if and only if m  …1 ÿ 2"†n < m ‡ 1. This implies that  m  n  2m ‡ 1 and "n  max 0; …n ÿ m ÿ 1†=2 : Thus, using the estimate n o jsinc xj  min 1; jxjÿ1 

2 1 ‡ j xj

…x 2 R†;

we ®nd that X tn 2Im

sinc2 …"nh† 

2m‡1 X

sinc2 …"nh†

nˆm

2‡

2m‡1 X nˆm‡2 ‰1

1; that the right-hand side of (26) converges absolutely and uniformly on compact subsets of C; and represents a function which is everywhere di€erentiable. Furthermore, (26) may be used to conclude that jF …x ‡ iy†j  Mejyj with some M > 0:

for x; y 2 R;

348

Higgins, Schmeisser, and Voss

We still have to show that F 2 Lp …R†: In view of statement (iv) of Theorem 2 and the representation (25), it is enough to prove that g : x 7ÿ!

X

f…2n† sinc2 …x ÿ n ‡ i†

n2Z

belongs to Lp …R†: A convexity argument, as we have used it in the proof of statements (i) and (iv) of Theorem 2, leads us to X sinc2 …x ÿ n ‡ i†

p

jg…x†j 

n2Z

!pÿ1


X

jf…2n†jp
sinc2 …x ÿ n ‡ i† :

n2Z

The ®rst factor on the right-hand side is a periodic function of x. As such, it is bounded on R by a constant C, say. Hence, for any A < B, we ®nd in view of statement (ii) of Theorem 2 that …B A

jg…x†jp dx  C


X n2Z

C

jf…2n†jp

…B A

sinc2 …x ÿ n ‡ i† dx

…1 ‡ 2†p kf kpp
ksinc…
‡ i†k22 : 2

This implies that g 2 Lp …R†: The proof is complete.

&

The equivalence between CSF and HSF holds in a much more general sense, including non-uniform sample points; see [17]. 4.

CAUCHY'S AND POISSON'S FORMULAE AND THE SAMPLING THEOREM

In this section we consider implications between the classical sampling formula CSF stated in Theorem 1 and the following two formulae for suitable functions f. Cauchy's Integral Formula (CIF). Let C be a simple, closed, recti®able, positively oriented curve in C. Then ( … f…z†; z 2 int C; 1 f…† d ˆ …27† 2i C  ÿ z 0; z 2 ext C; where int C denotes the interior of C and ext C denotes the exterior of C.

The Sampling Theorem and Several Equivalent Results in Analysis

Poisson's Summation FormulaÐA Special Case(PSF). … X 2n p ˆ 2wf ^ …0† ˆ w f f…t† dt; 2 w R n2Z

349

…28†

the series being absolutely convergent. We shall also need the following: Weak Identity Principle (WIP). If a member of B 2 vanishes on R, then it vanishes throughout C. We use again the notation CIF…Bp † ˆ) CSF…Bp † etc. introduced in § 2 and write ``‡'' to mean that two statements are connected by a logical ``and''. Theorem 4. CIF…B1w † () CSF…B1w † () PSF…B1w † ‡ WIP: (1) Proof of CIF(B1pw ) () CSF(B1pw ). See [4, pp. 527±528].

&

In the following proofs, we take w ˆ 1 since the results can be scaled to incorporate w 6ˆ 1: (2) Proof of CSF(B1p ) ˆ) PSF(B1p )+WIP. Let f 2 B 1 : Since we have shown in § 3 that CSF…B 1 † () DSF…B 1 †; we may represent f by formula (20) with w ˆ 1 and obtain z  X f…2n† sinc2 ÿ n f…z† ˆ 2 n2Z ‡ Now

…1 ÿ1

z  z 2X …ÿ1†n f 0 …2n† sin sinc ÿ n :  n2Z 2 2

…29†

…1 x  sinc2 ÿ n dx ˆ 2 sinc2 t dt ˆ 2 2 ÿ1

(see § 5) and …1 …1 x  x sinc ÿ n dx ˆ 2…ÿ1†n sin sin…t† sinc t dt ˆ 0 2 2 ÿ1 ÿ1 since the integrand of the last integral is an odd function. Hence, if we integrated both sides of (29) over R and interchanged summation and integration on the right-hand side, we would obtain (28) immediately. For a rigorous proof, we proceed as follows.

350

Higgins, Schmeisser, and Voss

First we integrate (29) over a ®nite interval ‰ÿA; AŠ. Then the uniform convergence of the series on compact subsets allows us to interchange summation and integration on the right-hand side. Therefore, lim

…A

A!1

ÿA

f…x† dx ˆ lim

A!1

‡ 

X n2Z

f…2n†

…A

x  sinc2 ÿ n dx 2 ÿA

X 2 lim …ÿ1†n f 0 …2n†  A!1 n2Z …A ÿA

sin

x  x ÿ n dx : sinc 2 2

…30†

The P integral in the ®rst series on the right-hand side is uniformly bounded and n2Z f…2n† converges absolutely, as a consequence of statement (ii) of Theorem 2. Hence the limit may be taken inside the summation. For a discussion of the second integral on the right-hand side, we consider … … A A=2ÿn x  x sinc ÿ n dx ˆ 2 sin sin…t† sinc t dt In …A† :ˆ 2 2 ÿA ÿA=2ÿn for a positive n: It is easily seen that In …A† is largest when A ˆ 2n. Hence … ÿ1 …0 sin…t† sinc t dt ‡ 2 sin…t† sinc t dt In …A†  In …2n†  2 ÿ2n

… ÿ1

ÿ1

2 

ˆ


2ÿ 2 ‡ lnj2nj : 

ÿ2n

dt ‡2 jtj

…0



ÿ1

jsin…t†j dt

The last bound holds for negative n as well. Therefore, if jf 0 …2n†j ˆ O…jnjÿ † as n ! 1; with > 1;

…31†

then the second series in (30), too, has an absolutely convergent majorant, and so we may take the limit inside the summation. Hence we have proved (28) for those functions f 2 B 1 which satisfy (31). Now let f be an arbitrary function in B 1 : Taking an " 2 …0; 1=4Š; we consider ÿ
f" …z† :ˆ f …1 ÿ 2"†z sinc2 …"z†:

The Sampling Theorem and Several Equivalent Results in Analysis

351

Then f" 2 B 1 and in addition it satis®es (31). Hence, by what we have proved already, (28) holds for f" . But then Lemma 1, which is a consequence  of CSF(B 2 ), assures that (28) holds for f as well. This completes the proof of PSF…B 1 †. Finally, let f be a member of B 2 such that f…x† ˆ 0 for all x 2 R: Then z z sinc2 …z 2 C† g : z 7ÿ!f 2 4 belongs to B 1 and vanishes on R: Applying CSF…B 1 † to g, we ®nd that g is identically zero. This implies that f vanishes throughout C: Hence the WIP holds. & (3) Proof of PSF(B1p )+WIP ˆ) CSF(B1p ): Let f 2 B1 . We wish to apply PSF…B1 † to the functions             sinc x ÿ and f2 :  7ÿ!f x ÿ sinc : f1 :  7ÿ! f 2 2 2 2 Therefore it must be con®rmed that they both belong to B1 for ®xed x 2 R. As to f1 , we have jf1 …t†j  jf…t=2†j for t 2 R and jf1 …†j  C ej= j for  2 C: Thus membership of B 1 is assured. Similarly for f2 . By applying Poisson's formula (28) with w ˆ 1 to f1 and f2 in turn, we obtain     … X t t sinc x ÿ dt …32† f…n† sinc…x ÿ n† ˆ f 2 2 2 R n2Z and 2

X

… f…x ÿ n† sinc…n† ˆ

n2Z

R

    t t f xÿ sinc dt : 2 2

…33†

By a change of variables, the right-hand sides of (32) and (33) are easily seen to be equal. Moreover, the left-hand side of (33) is equal to 2f…x†: Hence (32) and (33) imply that X f…n† sinc…x ÿ n† ˆ f…x† …x 2 R†: …34† n2Z

We now extend this sampling formula to C. Applying PSF…B 1 † to f and to f…
P‡ 1†, we ®nd in view of the absolute convergence of the series in (28) that n2Z jf…n†j < 1: Since the sinc function belongs to B 2 ; we obtain from (1) that jsinc…z ÿ n†j  M ej=zj

…z 2 C†:

352

Higgins, Schmeisser, and Voss

Hence X jf…n† sinc…z ÿ n†j  C ej=zj

…z 2 C†

…35†

n2Z

with some constant C: This shows that the series on the left-hand side of (34) converges absolutely and uniformly on compact subsets of C when x is replaced by z 2 C: Moreover, (34) and (35) imply that the function g : z 7ÿ!

X

f…n† sinc…z ÿ n†

…z 2 C†

n2Z

belongs to B 1 and coincides with f on the real line. Therefore, f ÿ g is a member of B 1 and vanishes on R. Bearing in mind that B1  B2 , we conclude by using the WIP that f ÿ g vanishes throughout C: This completes the proof. & One often needs to extend results in the present section to more general classes of functions. Using the kind of extension procedure that has now become familiar in our work so far, we can obtain the following Theorem 5; the special case p ˆ 2 will be needed in § 5. Theorem 5. CSF…Bp † ˆ) CIF…Bp †, Proof. Let f 2 B p . Then ÿ
f" : z 7ÿ!f …1 ÿ 2"†z sinc2 …"z†

1  p < 1. ÿ

" 2 …0; 1=4Š; z 2 C




belongs to B1 and we can apply the case p ˆ 1 of the theorem (taken from [4, p. 528] as cited above). That is, if CSF holds for f" , then CIF also holds for f" . We can now pass to the limit as " ! 0 in both CSF and CIF and ®nd & that, if CSF holds for f 2 Bp , then CIF also holds for this f. 5.

CAUCHY'S INTEGRAL FORMULA, THE SAMPLING THEOREM, AND SOME EQUIVALENT RESULTS IN FOURIER ANALYSIS

In this section we prove the equivalence of the following ®ve results, A, b 2 satisfy the B, C, D, and E. From now on we assume that members of B  Weak Identity Principle (WIP). A. Cauchy's Integral FormulaÐA Special Case. Let C be a simple, b 2 for some closed, recti®able, positively oriented curve in C . Let f 2 B 

The Sampling Theorem and Several Equivalent Results in Analysis

353

 > 0. Then,

1 2i

… C

f…† d ˆ ÿz

(

f…z†;

z 2 int C;

0;

z 2 ext C;

…36†

where int C denotes the interior of C and ext C denotes the exterior of C. B. A Modi®ed Gaussian Integral. …1 p 2 G…x† :ˆ eÿ…uÿix† du ˆ  ÿ1

The Hermite polynomials are given by  n d 2 2 eÿx ; Hn …x† :ˆ …ÿ1†n ex dx

…x 2 R†:

…37†

n 2 N0 :

…38†

These polynomials are orthogonal in the sense of L2 …R† with respect to the 2 weight eÿx , indeed the orthogonal weighted Hermite polynomials are given by   1 1 2 exp ÿ x Hn …x†: …39† n …x† :ˆ p 2 …  2n n!†1=2 C. Properties of the Orthogonal Hermite Polynomials Hn . C1. Mehler's formula for the Hermite polynomials: If jtj < 1, 1 ÿ…x2 ‡y2 †=2 X e nˆ0

2n n!

tn Hn …x†Hn …y†

( ) 1 x2 ÿ y2 …x ÿ yt†2 ÿ : ˆ p exp 2 1 ÿ t2 1 ÿ t2

…40†

C2. The set f n …x†g, np2 N0 , forms an orthonormal basis for L2 …R†. C3. Put 'n …x† ˆ …2n n! †1=2 n …x†. Then the weighted Hermite polynomials are eigenfunctions of the Fourier transform in that ' ^n ˆ in 'n . D. Plancherel's Theorem. D1. The Fourier transform F is a unitary operator on L2 …R†. D2. …F ÿ1 f†…t† is given by …F f†…ÿt†.

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b 2 . Then E. The Sampling Theorem. Let f 2 B  X f…n† sinc…t ÿ n†; f…t† ˆ

…41†

n2Z

the convergence being absolute and uniform on compact subsets of R. The equivalence of these ®ve results will be proved by showing that A ) B ) C ) D ) E ) A: In Theorem 6 we prove that A ) B: In Theorem 7 we prove that B ) C, by proving in part (i) that B ) C1 ) C2, and in part (ii) that B ) C3. In Theorem 8 we prove that C ) D, by proving that C2 and C3 ) D1 ) D2: In Theorem 9 we prove that D ) E. In Theorem 10 we prove that E ) A. Several parts of these proofs are to be found in Titchmarsh [20, Ch. 3]. There is much more detail in [20] than we need to record here, so we shall be content with brief outlines of this material. Before we can proceed to the proofs some preliminaries will be needed. 5.1. Preliminary Results and Lemmas. We will need some special integrals. First, the Gaussian integral …1 p 2 eÿu du ˆ : …42† ÿ1

The well-known real-variable proof of this famous formula4 proceeds by multiplying two copies of the integral together and passing to polar coordinates; then the integrand becomes directly integrable (e.g., [10, p. 496]). We will need the two integrals … … 3w 2 sinc …wt† dt ˆ sinc4 …wt† dt ˆ 1; …43† w 2 R R

and we also need the discontinuous integral … 1 1 sin…x† dx ˆ sgn :  ÿ1 x

…44†

Each of these three integrals has a straightforward real variable proof; see, e.g., [3, pp. 16±18]. For the proof of the main result we will need the following lemmas. 4

Usually associated with the name of Gauss, this integral was introduced by Abraham de Moivre in a privately printed pamphlet of 1733, entitled Approximatio ad summam terminorum binomii …a ‡ b†n in seriem expansi (Gauss was not born until 1777!). For more information and references see, e.g., C. B. Boyer and U. C. Merzbach, A History of Mathematics, second edition, John Wiley and Sons, 1989, p. 475.

The Sampling Theorem and Several Equivalent Results in Analysis

355

Lemma 3. Let C be a simple, closed, recti®able, positively oriented b 2 , then curve in C and let a 2 Int C. Let …z ÿ a†f…z† 2 B  ‡ f…† d ˆ 0: C

Proof. Apply Cauchy's formula A to …z ÿ a†f…z†. Thus, ‡ 1 … ÿ a†f…† d; z 2 Int C: …z ÿ a†f…z† ˆ 2i C ÿz The result is obtained when z ˆ a.

&

Although the following lemma can be seen as a special case of a more general theory of ``evaluation''-type kernels (e.g., [20, pp. 28±29; 3, § 6]), it is appropriate to give here a simple direct proof. Lemma 4. For z 2 C and w > 0, let us de®ne  p 3  2 jw …z† :ˆ 2 w …t ÿ ix†eÿ…tÿix†  sinc4 …wt† …z†; 2

…45†

where  denotes convolution. Then, when z ˆ u is real, and for all real x, 2

lim jw …u† ˆ F …x; u† :ˆ …u ÿ ix†eÿ…uÿix† :

w!1

…46†

Proof. For clarity we will usually suppress the dependence of F on x and write F …u†. Suppose ®rst that u ˆ 0. Then, using the two special integrals in (43) and the mean-value theorem, we have jjw …0† ÿ F …0†j … 1 3 4 F …t† w sinc …wt† dt ÿ F …0† ˆ 2 ÿ1 …1 3 jw sinc4 …wt†‰F …t† ÿ F …0†Šj dt  2 ÿ1 …1 3 0 maxjF …v†j jsinc3 …wt†j dt  2 v2R ÿ1 …1 3 0 maxjF …v†j sinc2 …wt† dt  2 v2R ÿ1 ˆ

3 maxjF 0 …v†j; 2w v2R

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and this vanishes as w ! 1. This is the essence of the matter; the proof for u 6ˆ 0 is not substantially di€erent. & b2 ; for all w > 0: Lemma 5. Let a 2 C, then J…z† :ˆ …z ÿ a†jw …z† 2 B 4w Proof. Throughout this proof c will denote various constants. First we show that J…z† has a representation of the form (2). From the de®nitions in Lemma 4, …1 ÿ
F …t† sinc4 w…z ÿ t† dt: …47† J…z† ˆ c…z ÿ a†w ÿ1

Now ÿ
1 sinc4 w…z ÿ t† ˆ p 2

… 4w ÿ4w

H…† i…zÿt† e d; w

…48†

where 8     1 jj 3 4 jj 2 > > > 2ÿ ÿ 1ÿ ; > > 1 jj 3 > > : 2ÿ ; 6 2w

jj  2w; 2w  jj  4w:

This formula (well known in the theory of B-splines, see, e.g., the article by Butzer and Stens in [13, p. 164]) can be established using just two integrations by parts in (48). Hence, from (47), J…z† ˆ c…z ÿ a† ˆ c…z ÿ a† ˆ c…z ÿ a†

…1 ÿ1

F …t†

… 4w

… 4w … 1 ÿ4w

… 4w ÿ4w

ÿ1

ÿ4w

H…†ei…zÿt† d dt

F …t† eÿit dtH…† eiz d

…†H…† eiz d;

…49†

say, the interchange being justi®ed by Fubini's theorem, since, as a function of t and ; jF …t†H…† ei…zÿt† j is integrable over R  ‰ÿ4w; 4wŠ.

The Sampling Theorem and Several Equivalent Results in Analysis

357

Let us write (49) in the form c…z ÿ a†J1 …z†. Now aJ1 …z† is already of the form (2); as for zJ1 …z†, an integration by parts in the formula for J1 …z† yields … 4w f…†H…†g0 eiz d; zJ1 …z† ˆ i ÿ4w

and so zJ1 …z† is also of the form (2). To show that J 2 L2 …R† we take z to be real, and integrate by parts a second time to obtain, via (49), … c…z ÿ a† 4w f…†H…†g00 eiz d; …50† J…z† ˆ z2 ÿ4w which clearly belongs to L2 …R† for real z. 5.2.

&

The Equivalences and Their Proofs.

Theorem 6. Cauchy's theorem A implies the modi®ed Gaussian integral B. Proof. We note ®rst that, because of (42), the statement (37) is correct when x ˆ 0. Thus, it must be shown that G…x† is independent of x, that is, …1 2 …u ÿ ix† eÿ…uÿix† du ˆ 0: …51† G0 …x† ˆ 2i ÿ1

Had the integrand in (51) satis®ed the conditions of Lemma 3, we could have obtained this result immediately. Since that is not the case we are going to approximate this integrand with jw …z† in (45), then by Lemma 5 we can indeed apply Lemma 3. As a consequence of Lemma 4, this will lead to …1 0 jw …u† du: …52† G …x† ˆ lim 2i w!1

ÿ1

Since jw …z† ˆ cJ…z†=…z ÿ a†, we ®nd from (50) that … c 4w f…†H…†g00 eiz d jw …z† ˆ 2 z ÿ4w … 4w … 0  c ‡ ˆ 2 f…†H…†g00 eiz d z 0 ÿ4w :ˆ A…z† ‡ B…z† say, where A…z† corresponds to the positive range of integration, B…z† to the negative. Now by Lemma 5, jw …z† satis®es the hypotheses of Lemma 3,

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and the same method of proof shows that the same is true of A…z† and B…z†, so that both integrals in the following vanish separately. Thus we have ‡ ‡ A…z† dz ÿ B…z† dz ˆ 0; …53† C0

C

where C is the contour consisting of the two parts ( C1 :ˆ fu : ÿR  u  Rg;

…54†

C2 :ˆ fz ˆ R ei : 0    g; and where C 0 is the contour consisting of the two parts 8 0 < C1 :ˆ fu : R  u  ÿRg;

…55† : C 0 :ˆ fz ˆ R ei : ÿ    0g: 2 „R 0 The contributions from C1 and C1 in (53) amount to ÿR jw …u† du, and when w ! 1 and R ! 1, this sum of contributions gives G 0 …x†=2i. „ So the proof will be completed by showing that C2 A…z† dz and „ 0 C2 B…z† dz both vanish as R ! 1, for all w > 0. Consider the ®rst of these integrals for large R; using (50) we have … … A…z† dz  R jA…R ei †j d C2

0

… … cR  4w i 00  2 f…†H…†g eiR e  d d R 0 0 … … c  4w jf…†H…†g00 j eÿR sin  d d ˆ R 0 0 … c 4w jf…†H…†g00 j d  R 0 ÿ! 0;

…R ! 1†; …w > 0†: 0

In a similar manner the integral of B…z† around C2 vanishes, and the proof is complete. & Theorem 7. (i) The modi®ed Gaussian integral B implies Mehler's formula C1, which in turn implies the property C2, that f n g; n 2 N0 forms an orthonormal basis for L2 …R†.

The Sampling Theorem and Several Equivalent Results in Analysis

359

(ii) The modi®ed Gaussian integral B implies that the weighted Hermite polynomials are eigenfunctions of the Fourier transform. Proof. (i) To prove that B ) C1. The proof is that in [20, Thm. 53, p. 77]. The integral (37) is used to modify each Hn on the left-hand side of (40), and standard manipulations then lead to the right-hand side. The integral (37) is used again to obtain the ®nal form of the right-hand side. No side results are needed. To prove that C1 ) C2, we adopt the orthogonality of f n g as a side result. This can be found in [20, § 3.5]) and is a simple consequence of a second order di€erential equation, obtained merely by di€erentiating the de®nition. The normalizing factors are found in [20, Thm. 54], by putting x ˆ y in Mehler's formula and integrating the result. The completeness follows from [20, Thm. 55, p. 79]. Titchmarsh proves the expansion property 2 … 1 n X am m …x† dx ˆ 0 …56† lim f…x† ÿ n!1 ÿ1 mˆ0 for f 2 L2 , where

am :ˆ

…1 ÿ1

f…x†

m …x†

dx;

m 2 N0 :

…57†

This is proved by noting that the ``kernel'' consisting of the right-hand side of Mehler's formula (40) satis®es a Dirac-Delta-like evaluation property as a consequence of [20, Thm. 17], needed here as a side result. The proof of [20, Thm. 17] only requires the integral (42) and standard procedures. This leads quickly to 1 X nˆ0

a2n

ˆ

…1 ÿ1

‰f…x†Š2 dx

…58†

for any f that belongs to L2 …R† \ C…R† and vanishes outside a bounded interval (we are clearly moving towards a Parseval completeness relation). The proof of (56) is completed by the usual density argument, with no further need for side results.

360

Higgins, Schmeisser, and Voss

(ii) The proof that B ) C3, that is, '^n ˆ in 'n is in [20, Thm. 57, p. 81]; aside from standard methods of analysis it uses only the modi®ed Gaussian integral (37). & Theorem 8. The orthonormal basis property of the weighted Hermite polynomials f n g; n 2 N0 , implies Plancherel's theorem. Proof. First we prove that C2 and C3 imply D1. The proof is that in [20, § 3.9], with an added comment. It requires the discontinuous integral (44), the formula (58) from the previous theorem and a side result, the Riesz±Fischer theorem for f n g in L2 …R†; this is of course a purely Hilbert space result, depending only on the orthonormal property of f n g and the fact that L2 is a complete metric space. However, Titchmarsh gives a direct derivation [20, Thm. 56, p. 80], using C2 of course, the orthonormal property of f n g. The proof, [20, pp. 82±83], is for even members of L2 …R† (essentially the same proof works for odd members, hence for all real valued members of L2 …R† with an obvious extension to complex valued members of L2 …R††. It is in two parts. First, it is noted that all ``Hermite coecients'' an , as in (57), with odd subscripts are zero; and as in (58) we have the Parseval-type relation 1 X nˆ0

a22n

ˆ2

…1 0

‰f…x†Š2 dx:

…59†

Now consider a new set of coecients f…ÿ1†n a2n g, and associate with it an even function g 2 L2 …R† by the Riesz±Fischer theorem (above). Then (59) holds with g in place of f, and this shows that the norms of g and f are equal. Second, it is to be shown that f and g are each other's Fourier cosine transform. It is worth commenting that an interesting feature of the proof, not explicitly mentioned by Titchmarsh, is that it also contains the one-toone property of the Fourier cosine transform. It is true that Titchmarsh remarks here that ``the relationship between f and g is plainly reciprocal.'' But this would be true of any other g, constructed in the same way but now choosing some distribution of signs other than f…ÿ1†n g. This new g would again have norm equal to that of f. Hence, this second part of the proof must use the particular distribution f…ÿ1†n g of signs. Indeed it does; the ^ factors …ÿ1†n arise from considering the special case 2n ˆ n ^ n …ÿ1† 2n of 'n ˆ i 'n (above).

The Sampling Theorem and Several Equivalent Results in Analysis

Titchmarsh uses this to give the calculation (r … …1 …1 sin…xy† sin…xy† 2 1 …ÿ1†n n …y† dy ˆ y y  0 0 0 r … 1 2 …ÿ1†n ˆ  0 r … x 2 …ÿ1†n ˆ  0

2n …t†

2n …t†

…1 0

361

) 2n …t†

cos…yt† dt dy

sin…xy† cos…yt† dy dt y

dt;

…60†

thus establishing the relationship …1 0

sin…xy† dy ˆ g…y† y

r … x 2 f…t† dt  0

in the special case where g…y† ˆ 2n …y† and where f…t† ˆ ^2n …t†. This last relationship is equivalent [20, p. 70] to g ˆ f ^ in the L2 -sense, and is ®nally and simply achieved from (60) by the completeness of f 2n g for even members of L2 …R†. The discontinuous integral used in the calculations above can be evaluated from (44). The implication D1) D2 is a standard proof from Fourier analysis (see, e.g., [18, Thm. 24, p. 17]). Initially the proof is for f 2 L1 …R† \ L2 …R†, and uses the uniqueness of the inverse transform and the fact that F preserves inner products; this is equivalent to the preservation of norms, via the standard polarization identity. The result is now extended to all of L2 …R† by the usual density argument. & Theorem 9. Plancherel's theorem D implies the sampling theorem E for b 2: f2B  Proof. The proof requires two side results; Schwarz inequality (a standard Hilbert space result whose proof requires nothing more than the linearity and Hermitian symmetry of the inner product) and the fact that feÿinx g; n 2 Z, is an orthogonal basis for L2 …ÿ; †. The orthogonality is a trivial calculation, and the completeness follows from the Weierstrass approximation theorem (e.g., [9, p. 66]). Now by Plancherel's theorem we have ÿ
f…t† ˆ F ÿ1 F f …t† ˆ …F F f †…ÿt†

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Higgins, Schmeisser, and Voss

in the norm of L2 …R†. That is, 2 … …A … ^ ixt dt ˆ f…t† ÿ f …x† e dx lim A!1 R

ÿA

R

2 … ^ ixt f…t† ÿ f …x† e dx dt ÿ

ˆ 0: It follows that f…t† ˆ

… ÿ

f ^ …x† eixt dx

a.e.

…61†

But f is continuous, hence this last relationship holds for all t 2 R. Now let fan g; n 2 Z, be a set of coecients which is to be determined, and using (61) and Schwarz's inequality let us consider … …  N N X X an sinc…t ÿ n† ˆ f ^ …x† eixt dx ÿ an ei…tÿn†x dx f…t† ÿ ÿ ÿ nˆÿN nˆÿN N X jeixt j f ^ …x† ÿ an eÿinx dx  ÿ nˆÿN …



N X p ^ an eÿin
kL2 …ÿ;† : 2 kf …
† ÿ

…62†

nˆÿN

Because of the orthogonal basis property of feÿinx g there are coecients fan g for which this last expression vanishes as N ! 1, and so from (62), f…t† ˆ

1 X nˆÿ1

an sinc…t ÿ n†;

uniformly on compact subsets of R. Finally, by the interpolatory property of the sinc function (namely the & formula sinc…m ÿ n† ˆ mn †, we ®nd that an ˆ f…n†; n 2 Z: A ®nal theorem closes our chain of implications. Theorem 10. The Sampling Theorem E implies the Cauchy integral b 2. formula A for f 2 B  Proof. If CSF holds on R for f 2 B 2 , then just as in the proof of Theorem 4, but with B 1 replaced with B 2 , and bearing the WIP in mind, we

The Sampling Theorem and Several Equivalent Results in Analysis

363

can extend CSF to C so that CSF…B 2 † holds. To complete the proof we b 2  B 2 , and Theorem 5 applies with p ˆ 2. & recall that B  

6.

THE SAMPLING THEOREM AND SOME EQUIVALENT RESULTS IN COMPLEX ANALYSIS

In this section we discuss connections between the sampling theorem and some fundamental theorems in complex analysis, namely Cauchy's integral formula, the maximum modulus principle, and the PhragmeÂn±LindeloÈf principle. While the sampling theorem applies to entire functions of exponential type with restricted growth on the real line, the fundamental results in complex analysis show their full power only when they are stated for a much wider class, commonly known as the analytic or holomorphic functions. In order to reach this class from the sampling theorem without involving already non-trivial complex analysis, we proceed in two steps as follows. b 2 introduced in § 2. First we extend the class B  De®nition 3. Let E be the class of all functions f for which there exist a b 2 , an integer n  0, and a polynomial p of degree at  > 0, a function g 2 B  most n ÿ 1 such that f…z† ˆ p…z† ‡ zn g…z† :

…63†

b 2 , the class E includes the polySince the null function belongs to B  nomials. Using standard properties of Lebesgue integrals (e.g., [16, § 26]), we easily verify the following assertions: b 2 such that f…z† ˆ b 2 there exist a c 2 C and a g 2 B (i) To f 2 B   iz c ‡ zg…z†: „  izt…Hint: Integration by parts and the formula e ˆ 1 ‡ iz 0 e dt:† b 2 and g 2 B b2 , then fg 2 B b2 . (Hint: Convolution.) (ii) If f 2 B   ‡ Used repeatedly, these statements allow us to conclude that, if f; g 2 E, then f ÿ g 2 E and fg 2 E. It follows that algebraically E has the structure of a ring. Admitting tools from complex analysis and the Paley±Wiener theorem, we can identify E as the class of all entire functions of exponential type with polynomial growth on the real line. In fact, if f is an entire function of exponential type  such that jf…x†j ˆ O…jxjk † as x ! 1, then f has a

364

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power series

P1

ˆ0

a z which converges everywhere. Moreover, g…z† :ˆ

f…z† ÿ

Pk

ˆ0 zk‡1

a z

is an entire function of exponential type  whose restriction to R belongs to b 2 . This shows that L2 . Now the Paley±Wiener theorem guarantees that g 2 B  f is of the form (63) with n ˆ k ‡ 1. Conversely, it is easily seen that every function of the form (63) is an entire function of exponential type with polynomial growth on the real line. Next we extend the class E. De®nition 4. Let U be a region. e (i) Denote by H…U† the class of all functions f : U ! C such that, for every compact subset C of U, there exists a sequence of functions belonging to E and converging to f uniformly on C. (ii) Denote by H…U† the class of all continuous functions f : U ! C which allow a representation f…z† ˆ

g…z† sin…z†

for z 2 U n Z

e with g 2 H…U†. e Since E includes the polynomials, H…U† includes the limits of locally uniformly convergent sequences of polynomials, in particular, it includes e power series that converge on U. Obviously, H…U†  H…U† since g…z† ˆ

sin…z†g…z† sin…z†

for z 2 U n Z;

e and along with g the numerator also belongs to H…U†. Admitting tools from e complex analysis, it can be shown that H…U† is exactly the class of all e functions which are analytic on U and that H…U† ˆ H…U†. As regards the de®nition of H…U†, we point out that sin…z† is the nodal function of the sampling series, that is, the canonical product formed by the nodes. The set of all functions f 2 E which the sampling series maps to the null function is a principal ideal generated by sin…z†: By introducing H…U† as a separate class, we avoid the necessity of knowing that result. We shall prove the equivalence of the following four results, E (see also § 5), F (an extension of A of § 5), G, and H.

The Sampling Theorem and Several Equivalent Results in Analysis

365

b 2 : Then E. The Sampling Theorem. Let f 2 B  X f…n† sinc…t ÿ n† f…t† ˆ

…64†

n2Z

the convergence being uniform on compact subsets of R. F. Cauchy's Integral Formula. Let U be a region, let C be a simple, e closed, recti®able, positively oriented curve in U, and let f 2 H…U†. Then ( … f…z†; z 2 int C ; 1 f…† …65† d ˆ 2i C  ÿ z 0; z 2 ext C : G. The Maximum Modulus Principle. Let U be a region and let  2 H…U†. Suppose that there exists a point z0 2 U such that j…z†j  j…z0 †j for all z 2 U. Then  is a constant. H. The PhragmeÂn±LindeloÈf Principle. Let U be an open sector with vertex at c 2 C and aperture = , where > 1=2. Let  be a function from H…U† having a continuous extension on U. Suppose that j…z†j  M on the two rays of the boundary of U and that, for some increasing sequence …Rn †n2N of positive real numbers with limn!1 Rn ˆ 1 and a 2 …0; †, we have ÿ
as n ! 1 j…c ‡ Rn ei' †j ˆ O exp…R n † uniformly in ' for c ‡ Rn ei' 2 U. Then j…z†j  M throughout U: We shall show the following implications: E ˆ)F ˆ)G ˆ)H ˆ)E: For the ®rst, the following statement will be useful. b 2 and let h…z† ˆ sin…cz† with c > 0: Then gh 2 Lemma 6. Let g 2 B 

b2 : B ‡c

„ Proof. Since g has a representation g…z† ˆ ÿ eizt …t† dt, we obtain … eicz ÿ eÿicz  izt e …t† dt g…z†h…z† ˆ 2i ÿ … ‡c  … ÿc 1 izt izt e …t ÿ c† dt ÿ e …t ‡ c† dt ; ˆ 2i ÿ‡c ÿÿc b2 . which shows that gh 2 B ‡c

&

366

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The Sampling Theorem E Implies Cauchy's Integral Formula F. e Let f 2 H…U†. Then f ˆ limj!1 fj with fj 2 E. By repeated application of Lemma 6, we conclude that, for ®xed j, appropriately chosen k 2 N, and all " 2 (0,1), there exists a  > 0 so that hj;" : z 7ÿ!fj …z† sinck …"z† b 2 : From the preceding section we know already that belongs to B  b 2 and so statement E implies Cauchy's integral formula for functions in B  for hj;" ; in particular. Letting " ! 0 and j ! 1 afterwards, we conclude that Cauchy's integral formula holds for the functions fj and the limiting function f as well. & Cauchy's Integral Formula F Implies the Maximum Modulus Principle G. Under the hypothesis of the maximum modulus principle,  is a continuous function with a representation …z† ˆ f…z†= sin…z† …z 2 U n Z†, e where f 2 H…U†: Now let  2 U: From Cauchy's integral formula, which is known to hold for f and the sine function, it follows easily (see [19, § 2.43, p. 83]) that there exist expansions f…z† ˆ

1 X

an …z ÿ †n

and

nˆ0

sin…z† ˆ

1 X

bn …z ÿ †n ;

nˆ0

the series converging uniformly in some disc D…† centered at . Let bk …k ˆ 0 or k ˆ 1† be the ®rst non-vanishing coecient of the second series. Then, in a concentric disc D1 …†  D…†; we have !ÿ1 1 X bk …z ÿ †k bn nÿk ˆ 1‡ …z ÿ † sin…z† b nˆk‡1 k ˆ

1 X

…ÿ1†

mˆ0

m

1 X bn …z ÿ †nÿk b k nˆk‡1

!m :

The right-hand side can be obtained as the limit of a sequence of polynomials gn …z† converging uniformly on D1 …†. Next we observe that an ˆ 0 for n < k as a consequence of the continuity of . Therefore, N X f…z† gM …z† ˆ lim lim an …z ÿ †nÿk ; sin…z† M!1 bk N!1 nˆk

The Sampling Theorem and Several Equivalent Results in Analysis

367

uniformly on D1 …†. This shows that for every  2 U, there exists a neighe  †. As a bourhood U such that the restriction of  to U belongs to H…U consequence, Cauchy's integral formula applies to  locally. Taking circles as paths of integration, we ®nd that for every  2 U, there exists a  > 0 so that …† ˆ

1 2

… 2 0

… ‡  eit † dt :

From these formulae the maximum modulus principle follows in an elementary way (see [19, § 5.11, p. 165, ®rst proof ]). & The Maximum Modulus Principle G Implies the PhragmeÂn±LindeloÈf Principle H. Performing an appropriate substitution of the form z 7! c ‡ ei' z; we may suppose that  U ˆ z 2 C : z 6ˆ 0; j arg zj < =…2 † ;

where > 1=2 :

For  2 … ; † and " > 0; consider ÿ
f" …z† :ˆ exp ÿ" exp… ln z† : The function f" is well de®ned on U by agreeing that =…ln z† ˆ arg z 2 …ÿ; †: For k 2 N; we have   1 X …ÿ1†nÿ1 z ÿ k n : ln z ˆ ln k ‡ k n nˆ1 The series converges locally uniformly in the disc fz 2 C : jz ÿ kj < kg. It follows that the sequence of polynomials   k2 X …ÿ1†nÿ1 z ÿ k n …k 2 N† gk …z† :ˆ ln k ‡ n k nˆ1 converges on U locally uniformly to ln z: Hence the logarithm belongs to e H…U†: Since the exponential function has a power series which converges e everywhere, we conclude that f" 2 H…U†: Next we observe by a simple calculation that   jf" …z†j ˆ exp ÿ"jzj cos… arg z† < 1

for z 2 U:

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Under the hypothesis on , it follows that " :ˆ f"  2 H…U† and that j" …Rn ei' †j ˆ

j…Rn ei' †j exp…Rn †

ÿ
exp ÿ"R n cos… arg z† ‡ R n ! 0

as n ! 1

uniformly for ' 2 ‰ÿ=…2 †; =…2 †Š: Now the maximum modulus principle implies that j" …z†j  M for z 2 U and every " > 0. Letting " ! 0; we obtain that j…z†j  M throughout U. & The PhragmeÂn±LindeloÈf Principle H Implies the Sampling Theorem E. b 2 and f…x† ˆ O…jxjÿ1 † as x ! 1; that is, First we suppose that f 2 B   b 2 \ B 2 : Then the series f2B   g…z† :ˆ

1 X

f…n† sinc…z ÿ n†

nˆÿ1

converges absolutely and uniformly on compact subsets of C and represents e a function g 2 H…U†: Now we consider …z† :ˆ

f…z† ÿ g…z† sin…z†

…z 62 Z†:

e The numerator f ÿ g belongs to H…U†: Near an integer m, we may write …m ‡ † ˆ …ÿ1†m

1 f…m ‡ † ÿ f…m† sinc  X …ÿ1†n ; ‡ 1 f…n† …m ‡  ÿ n† sin…† nˆÿ1 n6ˆm

where  2 C and 0 < jj < 1=2: As  ! 0; the ®rst term on the right-hand side converges to …ÿ1†m f 0 …m†=; the second term has a limit, trivially. Hence lim!0 …m ‡ † exists, and so  has an extension belonging to H…C†. b 2  B 2 , there exists a Next we estimate the growth of . Since f 2 B   constant M such that jf…z†j  M ej=zj

for z 2 C :

The Sampling Theorem and Several Equivalent Results in Analysis

369

Besides, jxf…x†j is bounded on R; by a constant C, say. In this situation the PhragmeÂn±LindeloÈf principle implies (see [1, § 6.2, Thm. 6.2.4 and its proof ]) that jzf…z†j  C ej=zj

for z 2 C :

…66†

Now, for k 2 N, we introduce the pair of straight lines 
k :ˆ z ˆ i…k ‡ 1=2† ‡ t exp… i=4† : t 2 R : They intersect at i…k ‡ 1=2† and cross the real line at …k ‡ 1=2†: Using (66) and the estimate j sin…z†j  …exp…j=zj† ÿ 1†=2; we easily see that f…z† < 1 and lim Ak ˆ 0: …67† Ak :ˆ sup k!1 z2
k sin…z† Recalling the de®nition of g, we may write 1 X g…z† …ÿ1†n : ˆ f…n† …z ÿ n† sin…z† nˆÿ1

A detailed discussion of the right-hand side (using HoÈlder's inequality and distinguishing the cases j=zj < k=2 and j=zj  k=2† yields that g…z† < 1 and lim Bk ˆ 0: …68† Bk :ˆ sup k!1 z2
k sin…z† Next we de®ne Rk;n :ˆ

q …k ‡ 1=2†2 ‡ …n ‡ 1=2†2

and note that i…k ‡ 1=2† ‡ Rk;n exp…i'†

…0  ' < 2†

cannot be a zero of sin…z†; at such points we even have jsin…z†j  1 when n is large. Hence, for ®xed k 2 N and > 1; we obtain   ÿ
 i…k ‡ 1=2† ‡ Rk;n ei' ˆ O exp…R † as n ! 1; …69† k;n uniformly for ' 2 ‰0; 2†: Now we argue as follows. The pair
k of straight lines decomposes the complex plane into four sectors. From (67)±(69) we see that the PhragmeÂn± LindeloÈf principle applies to  on each of these sectors. It yields that j…z†j  Ak ‡ Bk

370

Higgins, Schmeisser, and Voss

on each of them, and so on the whole of C: Letting k ! 1; we obtain that …z†  0; which implies that f…z†  g…z† on C: Thus we have proved that the b 2 as b 2 \ B 2 †; but then it holds for f 2 B sampling formula holds for f 2 B    well (see Lemma 2 and Remark 1.) &

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