The limiting behavior of certain sampling series and cardinal splines

Journal Pre-proof The limiting behavior of certain sampling series and cardinal splines W.R. Madych

PII: DOI: Reference:

S0021-9045(19)30095-4 https://doi.org/10.1016/j.jat.2019.105298 YJATH 105298

To appear in:

Journal of Approximation Theory

Received date : 27 March 2018 Revised date : 27 March 2019 Accepted date : 10 September 2019 Please cite this article as: W.R. Madych, The limiting behavior of certain sampling series and cardinal splines, Journal of Approximation Theory (2019), doi: https://doi.org/10.1016/j.jat.2019.105298. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2019 Elsevier Inc. All rights reserved. ⃝

Journal Pre-proof

The Limiting Behavior of Certain Sampling Series and Cardinal Splines W. R. Madych Department of Mathematics University of Connecticut, Storrs, CT 06269, USA

Abstract

1

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Many families of cardinal sampling kernels {Φα (x) : α ∈ I} consist of members that decay rapidly as x → ±∞ and enjoy the property that, as the parameter α tends to its limiting value, Φα (x) tends to the classical cardinal sine, P sinc(x). We study the limiting behavior of the sampling series n cn Φα (x − n) when the data sequence {cn } remains fixed. In the case when the sampling kernels are damped cardinal sines, appropriate conditions on the damping functions allow us to characterize the limiting behavior of the corresponding sampling series for a wide class of data sequences {cn }. These results lead to conclusions regarding the limiting behavior of classical piecewise polynomial cardinal splines when the fixed data sequence {cn } is of polynomial growth and the order tends to ∞. One application of this development shows that the classical spline summability method is regular. Another consequence, perhaps more interesting and unexpected, is a resolution of I. J. Schoenberg’s conjecture concerning the recovery, in terms of their samples {f (n)}, of members f (x) of the Bernstein class via the spline summability procedure.

Introduction Extended abstract

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1.1

A cardinal sampling series is the expression (1)

∞ X

n=−∞

cn Φ(ρx − n) 1

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where the sampling data {cn : n = 0, ±1, ±2, . . .} is a bi-infinite sequence of numerical values, the sampling kernel Φ(x) is a continuous function of the real variable x, −∞ < x < ∞ that satisfies ( 1 if n = 0 (2) Φ(n) = 0 if n = ±1, ±2, . . ., and the positive constant ρ is the sampling rate. If the series (1) converges locally uniformly it defines a continuous function f (x) with the property that f (n/ρ) = cn , n = 0, ±1, ±2, . . . . In this case (1) extends the discrete data
sequence {(n/ρ, cn ) : n = 0, ±1, ±2, . . .} to the continuous graph { x, f (x) : −∞ < x < ∞}. In this article we will be concerned with the case of fixed sampling rate ρ. Since a dilation argument shows that there is no loss of generality by restricting attention to the special case ρ = 1 we do so in what follows. The quintessential example of a sampling kernel is the familiar cardinal sine Φ(x) = sinc(x) :=

sin πx πx

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f

(3)

that plays a critical role in the Whittaker-Kotelnikov-Shannon sampling theorem that implies that f (x) =

∞ X

f (n) sinc(x − n)

rep

(4)

n=−∞

whenever f (x) is in the Paley-Wiener class P Wπ . Formulations of this result and further references can be found in [8, 14, 15, 17, 38, 40].

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The main subjects of our study are sampling series involving a family {Φα (x) : α ∈ I} of sampling kernels parametrized by α ∈ I where I is an index set with a limit point.

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For simplicity we take the limit point to be 0 and the index set I to be the interval (0, 1] or the sequence {1, 1/2, 1/3, . . .}, namely I = {α : 0 < α ≤ 1} or I = {α = 1/k : k = 1, 2, 3, . . .}.

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The families {Φα (x)} are such that its members enjoy faster decay than the cardinal sine sinc(x) as x → ±∞ but, in the limit as α → 0, they converge to sinc(x). It is known that for many such families of sampling kernels the corresponding sampling series converge to f (x) when f is in the Paley-Wiener class P Wπ and the data sequence {cn } consists of point samples of f , namely cn = f (n), n = 0, ±1, ±2, . . . ; for example, see [1, 2, 6, 13, 18, 22, 33]. The subject of this article concerns much wider classes of data sequences. 2

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An important example of such cardinal sampling series are Schoenberg’s classical piecewise polynomial cardinal splines, [33]. If the sequence of data samples {cn } grows no faster than a polynomial then the spline of order 2k with knots at the integers and of no greater than polynomial growth that interpolates {cn } is unique and can be expressed in terms of translates of a “fundamental spline” or sampling function Lk (x). Namely, if we use the notation Sk ({cn }, x) to denote this spline then Sk ({cn }, x) =

∞ X

n=−∞

cn Lk (x − n).

If we use the identification α = 1/k and Φα (x) = P Lk (x) then the series defining Sk ({cn }, x) is the cardinal sampling series n cn Φα (x − n). It is known that if f is in the Paley-Wiener class P Wπ then (5)

lim Sk ({f (n)}, x) = f (x).

k→∞

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f

More generally, (5) is valid for functions f in P Wπ∗ , the class of those functions f whose kth order derivative, f (k) , is in P Wπ for some k; see [34] for details. Such functions need not be bounded. Since P Wπ∗ contains P Wπ , I. J. Schoenberg [33, p. 106, Definition 2] proposed (5) as a summability method for the cardinal sine series (4).

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However, while it was known that that (4) is valid for a class of functions significantly wider than P Wπ , [14, 39], the question of regularity, in other words, the question of whether the convergence of (4) implies (5), was not addressed in [33]. This may be viewed as somewhat of a gap in the theory since, to my knowledge, this question has also not been addressed elsewhere, despite the fact that the subclasses of Eπ for which spline summability is known to be valid [12, 29, 30, 33–35] do not contain all convergent cardinal sine series [3].

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Schoenberg also studied the spline summability method for functions in the Bernstein class Bπ that consists of those entire functions of exponential type no greater than π that are bounded on the real axis. Not only is Bπ not contained in P Wπ∗ but the samples {f (n)} do not uniquely determine functions f in Bπ . Nevertheless, based on positive results for almost periodic members of Bπ and other considerations, Schoenberg conjectured that if f is in Bπ then with an appropriate constant c lim Sk ({f (n)}, x) = f (x) + c sin πx

k→∞

uniformly on compact subsets of the real axis, [35, Conjecture 1, p. 309]. 3

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In this article we settle the issues mentioned above. More precisely, we show that the following are true: (i) The spline summability method is regular, in the sense that the convergence of (4) implies (5). (ii) There are functions f in the Bernstein class Bπ such that Sk ({f (n)}, x) fails to converge for every non-integer x as k tends to ∞. Our development begins in Section 2 by considering families of sampling kernels {Φα (x)} consisting of damped cardinal sines and studying the limiting behavior as α → 0 of the corresponding sampling series. By damped cardinal sines we mean kernels of the form Φα (x) = sinc(x)φα (x) where φα (x) is relatively smooth, decays as x → ±∞, φα (0) = 1, and limα→0 φα (x) = 1.

1.2

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In Section 3 we show that the fundamental interpolating spline Lk (x) is essentially a damped cardinal sine. As a corollary, the results of section 2 can be re-formulated for classical piecewise polynomial cardinal splines. This leads to several unexpected results concerning splines, including items (i) and (ii) mentioned above.

Notation and conventions

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We use standard notation and only alert the reader to the fact that Eσ , where σ > 0, denotes the class of entire functions of exponential type no greater than σ that have no greater than polynomial growth along the real axis. In view of the distributional variant of the Paley-Wiener Theorem, for example see [16, Theorem 1.7.7], Eσ consists of the Fourier transforms of distributions with support in the interval [−σ, σ]. The Fourier transform of a tempered distribution u is denoted by u b and, in the normalization used here, is defined by Z u b(ξ) =

∞

e−iξx u(x)dx

−∞

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when u is an integrable function. As is customary, the symbol C is used to denote generic constants whose value depends on the context.

1.3

Contents

Here we provide a somewhat more extensive outline of the contents to supplement that given Subsection 1.1,

4

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In Section 2 we study the behavior of cardinal sampling series where the sampling kernel Φα is a dampled cardinal sine, namely Φα (x) = sinc(x) φα (x). Under appropriate restrictions on the damping functions we show the following: (T1) If the sequence of samples {cn } is of no greater that polynomial growth then as α → 0 the sampling series either converges to an entire function in Eπ uniformly on compact subsets of R or fails to converge on all but a finite number of non-integer values. (T2) If the sequence of samples {cn } are those of a convergent cardinal sine series f then as α → 0 the sampling series converges to f (x) uniformly on compact subsets of R. (T3) If the sequence of samples {cn } is such that cn tends to 0 as n tends to ±∞ and the sampling series converges for one non-integer value of x then corresponding cardinal sine series converges uniformly on compact subsets.

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In Section 3 we review some basic facts concerning piecewise polynomial cardinal splines and show that the fundamental functions Lk (x) are damped cardinal sines with damping functions that enjoy the restrictions necessary for items (T1)-(T3) mentioned above to be valid. As a result, it follows that the spline summability method is regular; such results are often referred to as Abel type or Abelian theorems. Furthermore, (T3) implies that a theorem of Tauberian type is valid for the cardinal spline summability method and allows us to conclude that Sk ({f (n}, x) fails to converge as k tends to ∞ for certain functions f in the Bernstein class Bπ .

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This development involves technical details that may make it tiresome to follow the thread. A succinct guide including more background and motivational material can be found in [26].

Damped cardinal sines

2.1

Setup

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2

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Random observations that are germane but not crucial to our development are briefly summarized in Section 4.

Consider the sampling series with coefficients {cn } = {cn : n = 0, ±1, ±2, . . .} Fα ({cn }, x) =

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(6)

∞ X

n=−∞

cn

sin π(x − n) φα (x − n) π(x − n)

where {φα (x) : α ∈ I} is a family of continuous functions on R with φα (0) = 1, parameterized by α ∈ I, chosen so that the series converges absolutely for each α. I is an index set with a limit point. The idea, of course, is that 5

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Fα ({cn }, x) interpolates the data, namely Fα ({cn }, m) = cm for all integers m, and, under appropriate conditions on the coefficients {cn } and the family {φα (x)}, converges to a function in Eπ as α tends to the limit point. The specific case of (6) with φα (x) = sinc(αx), α > 0, is reminiscent of the Bernstein-Boas formula for bounded functions in Eσ , σ < π, [19, (14), p. 160]. For simplicity we take the limit point to be 0 and the index set I to be the interval (0, 1] or the sequence {1/k : k is an integer ≥ k0 }, namely I = {α : 0 < α ≤ 1} or I = {α = 1/k : k ≥ k0 } where k0 is a positive integer. Furthermore, we assume that for all α in I (7)

φα (0) = 1

and φα (−x) = φα (x).

f

(8)

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In what follows, ν is a positive integer that is used to describe the following useful condition on the family {φα (x)}.

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Hypothesis H(ν): For each α in I the function φα (x) is in C ν (R) and for j = 0, 1, . . . , ν there is a constant C independent of x and α such that |φ(j) α (x)| ≤ C

αj . 1 + |αx|ν

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(9)

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Examples: Simple examples of families of functions {φα (x)} satisfying hypothesis H(ν) can be constructed from any function ψ(x) in C ν (R) that satisfies ψ(0) = 1, ψ(−x) = ψ(x), and |ψ (j) (x)| ≤

C , 1 + |x|ν

j = 0, 1, . . . , ν ,

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by setting φα (x) = ψ(αx). Specific examples include 2

ψ(x) = e−x ,


ν ψ(x) = sinc(x) ,

ψ(x) = 1 + x2


−ν/2

.

The following technical lemma provides an alternate expression for Fα ({cn }, x) in relation (6) that is useful in the development below. 6

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Lemma 1. Suppose cn = O(nρ ) as n → ±∞ where 0 < ρ < κ, κ is an integer ≥ 1, and the family {φα (x)} satisfies Hypothesis H(ν) with ν ≥ κ. Then, for all α > 0, the series (6) converges absolutely and Fα ({cn }, x) is a well defined function of x in C ν (R). Furthermore, Fα ({cn }, x) can be re-expressed as ∞ n o X cn κ (10) Fα ({cn }, x) = c0 sinc(x)φα (x) + x sinc(x − n)φ (x − n) α nκ n=−∞ n6=0

−

sin(πx) {Pα (x) + Rα (x)} π

where (11)

Pα (x) =

κ−1 X

Bα,m xm

m=0

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f

is a polynomial of degree κ − 1 whose coefficients are given by ( m ) ∞ X (−1)n cn X (−n)j (j) (12) Bα,m = φα (n) , m = 0, 1, . . . , κ − 1, nm+1 j! n=−∞ j=0 n6=0

and

lim Rα (x) = 0

rep

(13)

α→0

uniformly on compact subsets of R.

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Proof. Note that {cn /nκ } is in `p when (κ − ρ)p > 1. Hence, since ρ < κ, we may conclude that {cn /nκ } is in `p for sufficiently large p < ∞. It follows that the series G(x) =

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(14)

∞ X cn sinc(x − n) κ n n=−∞ n6=0

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converges absolutely and uniformly on R and, in view of the Plancherel-Polya theorem [19, p. 152], G(x) is a function in Eπ ∩ Lp (R) with G(0) = 0 and G(n) = cn /nκ when n = ±1, ±2, . . . . Now use the facts that sin π(x − n) = {sin πx}(−1)n and κ−1

X xm 1 xκ − = κ x−n n (x − n) m=0 nm+1 7

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to re-express Fα ({cn }, x) as (15)

Fα ({cn }, x) = c0 sinc(x)φα (x) + xκ Gα (x) −

sin(πx) Aα (x) π

where (16)

Gα (x) =

∞ X cn sinc(x − n)φα (x − n) κ n n=−∞ n6=0

and

) ( κ−1 X xm φα (x − n). Aα (x) = (−1) cn nm+1 n=−∞ m=0 ∞ X

n

n6=0

Aα (x) can be re-expressed as (17)

Aα (x) =

( κ−1 X

Bα,m xm

(18)

Bα,m

∞ X (−1)n cn = nm+1 n=−∞

( m X (−n)j j=0

j!

)

φ(j) α (n) ,

m = 0, 1, . . . , κ − 1,

rep

n6=0

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where

+ Rα (x)

f

m=0

)

and (19)

lim Rα (x) = 0 →0

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uniformly on compact subsets of R.

The conclusions of the Lemma follow from (15), (17), (18), and (19).

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It remains to verify (17), (18), and (19). To do so use the definition of Aα (x) to write

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(20) where

Aα (x) =

κ−1 X

Aα,j (x)

j=0

∞ X (−1)n cn Aα,j (x) = x φα (n − x). nj+1 n=−∞ j

n6=0

8

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To isolate the dependence on x of each of the terms Aα,j (x) use φα (x − n) = φα (n − x) and Taylor’s expansion (κ−j−1 ) X (−x)m (−x)κ−j (κ−j) φ(m) (n) + φ (n − x˜) φα (n − x) = m! α (κ − j)! α m=0 where x˜ is a point between 0 and x to write ( κ−j−1 ) ∞ n m X X (−1) c (−n) n xm (21) Aα,j (x) = xj φ(m) + A˜α,j (x) α (n) j+m+1 n m! n=−∞ m=0 n6=0

where A˜α,j (x) = xκ

∞ X (−1)n cn (−1)κ−j (κ−j) (n − x˜). φ nj+1 (κ − j)! α n=−∞ n6=0

To see that lim A˜α,j (x) = 0

(22)

f

α→0

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uniformly on compact subsets of R, use the fact that for sufficiently large n |φ(κ−j) (n − x˜)| ≤ C α

ακ−j 1 + |αn|κ

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and choose % such that ρ ≤ % and j < % < κ to write    X  X |A˜α,j (x)| ≤ C|x|κ |n|ρ−j−1 ακ−j + |n|ρ−j−1−κ α−j   |n|>1/α

0<|n|≤1/α

κ

.

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≤ C|x| α

κ−%

The last inequality implies (22).

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Substituting expression (21) for Aα,j (x) into (20) and rearranging terms results in (17) and (18) where Rα (x) =

κ−1 X j=0

(19) follows by virtue of (22). This completes the proof of Lemma 1

9

A˜α,j (x).

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Lemma 2. As α tends to 0 the polynomials {Pα (x), α ∈ I} in Lemma 1 either (a) converge uniformly on compact subsets of R to a polynomial P (x) of degree κ − 1 or (b) diverge for all but at most κ − 1 values of x. In other words, if Pα (x) converges for κ values of x as α tends to 0 then lim Pα (x) = P (x)

α→0

uniformly on compact subsets of R. Proof. If the polynomials Pα (x) converge for κ distinct values of x, say x1 , . . . , xκ , as α → 0 then the coefficients {Bm,α : m = 0, . . . , κ − 1}, being a solution to the Vandermonde system κ−1 X

Pα (xj ) =

Bα,m xm j

j = 1, . . . , κ ,

m=0

also converge. Namely, lim Bα,m = Bm ,

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It follows that (23)

m = 0, . . . , κ − 1.

f

α→0

lim Pα (x) =

α→0

κ−1 X

Bm xm = P (x)

rep

m=0

uniformly on compact subsets of R.

Hence, if Pα (x) diverges for one value of x as α → 0 then it must diverge for all but at most κ − 1 of values of x. The conclusions of Lemma 2 follow.

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Lemma 3. Suppose Fα ({cn }, x) and Pα (x) are defined as in Lemma 1. Then, when x is not an integer and α tends to 0, Fα ({cn }, x) converges if and only if Pα (x) converges.

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(24)

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Proof. Recall that the cardinal sine series G(x) defined by (14) converges absolutely and uniformly. Hence its mollification Gα (x) defined by (16) satisfies lim Gα (x) = G(x)

α→0

uniformly on R. In view of (24) and Lemma 1 when x is not an integer then, as α tends to 0, Fα ({cn }, x) converges if and only if Pα (x) converges. Lemmas 1, 2, and 3 lead to the following conclusion. 10

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Theorem 1. Suppose cn = O(nρ ) as n → ±∞ where 0 < ρ < κ, κ is an integer ≥ 1, and the family {φα (x)} satisfies Hypothesis H(ν) with ν ≥ κ. Then, for all α in I, the series (6) converges absolutely and Fα ({cn }, x) is a well defined function of x in C ν (R). Furthermore, as α tends to zero, Fα ({cn }, x) either (a) converges uniformly on compact subsets of R to a function F (x) in Eπ or (b) diverges for all but at most κ − 1 non-integer values of x. In case (a) the function F is defined by ∞ o n X cn sin(πx) κ sinc(x − n) − P (x) F (x) = c0 sinc(x) + x κ n π n=−∞ n6=0

where P (x) is a polynomial of degree ≤ κ − 1 determined by relation (23). An alternate expression for Bm,α Suppose {an } and {bn } are the even and odd parts, respectively, of the sequence {cn }. Namely cn + c−n 2

and bn =

cn − c−n . 2

f

an =

rep

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Then the coefficients Bm,α defined by (12) can be re-expressed as follows: ( m ) ∞ X (−1)n bn X (−n)j (j) Bm,α = 2 φα (n) if m is even, and m+1 n j! n=1 j=0 ( m ) (25) ∞ X (−1)n an X (−n)j (j) φα (n) Bm,α = 2 if m is odd. nm+1 j! n=1 j=0

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To see this, let

wn =

m X (−n)j j=0

j!

φ(j) α (n)

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and note that (12) can be expressed as        ∞ ∞ X    n n X (−1) bn  (−1) an (26) Bα,m = wn + wn . m+1 m+1       n=−∞ n n=−∞ n n6=0

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n6=0

If m is even then the series of terms in the first sum on the right hand side of (26) is odd and hence sum to 0 while the series of terms in the second sum on the right hand side of (26) is even. This yields the first identity in (25). Analogous reasoning when m is odd results in the second identity in (25).

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Theorem 2. Suppose the sequence {cn } consists of the coefficients of a convergent cardinal sine series f (x) =

∞ X

n=−∞

cn sinc(x − n)

and the family {φα (x), α ∈ I} satisfies Hypothesis H(ν) with ν ≥ 3. Then, for every α in I, the series (6) converges absolutely and Fα ({cn }, x) is a well defined function in C ν (R). Furthermore, lim Fα ({cn }, x) = f (x)

(27)

α→0

uniformly on compact subsets of R. Proof. Since {cn } are the coefficients of a convergent cardinal sine series, cn = o(n2 ) so we can apply Lemma 1 with κ = 3. Hence Fα (x) = c0 sinc(x)φα (x) + x3 Gα (x) −

f

where

sin(πx)  Pα (x) + Rα (x) π

∞ X cn sinc(x − n) 3 n n=−∞

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lim Gα (x) =

α→0

n6=0

uniformly on R,

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lim Rα (x) = 0

α→0

uniformly on compact subsets of R, and Pα (x) = Bα,0 + Bα,1 x + Bα,2 x2

Bα,m

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where

∞ X (−1)n cn = nm+1 n=−∞

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n6=0

( m X (−n)j j!

j=0

)

φ(j) α (n) ,

m = 0, 1, 2.

Also, since {cn } are the coefficients of a convergent cardinal sine series, both

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B0 =

∞ X (−1)n bn n=1

n

and B1 =

∞ X (−1)n an n=1

n2

converge, where {bn } and {an } are the odd and even parts of {cn }, [3, Theorem 1]. It follows that ∞ X (−1)n bn B2 = n3 n=1 12

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converges absolutely. In view of (25) we may conclude that (28)

lim Bα,m =

α→0

∞ X (−1)n cn = Bm , nm+1 n=−∞

m = 0, 1, 2, .

n6=0

Hence Pα (x) converges uniformly on compact subsets to B0 + B1 x + B2 x2 as α tends to 0 and ∞ n o X cn lim Fα ({cn }, x) = c0 sinc(x) + x3 sinc(x − n) α→0 n3 n=−∞ n6=0

  ∞ x sin(πx) X 1 x2 n + . − (−1) cn + π n n2 n3 n=−∞ n6=0

Since ∞ ∞ X sin(πx) X cn x3 (−1)n cn sinc(x − n) = n3 π n3 (x − n) n=−∞ n=−∞ n6=0 ∞ X

roo

n6=0

f

x3

sin(πx) (−1)n cn = π n=−∞ n6=0

1 1 x x2 + + 2+ 3 x−n n n n



,

rep

(27) follows.



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The conclusions of Theorem 2 can be taken as the basis of a summability method for the cardinal sine series. Namely, imitating Schoenberg [33, Definition 2, p. 106] leads to the following:

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Definition Suppose F = {φα (x) : α ∈ I} is a family of functions satisfying hypothesis H(ν) for some positive integer ν. Then the cardinal sine series ∞ X cn sinc(x − n) n=−∞

Jo u

is said to be F summable to the entire function F (x) in Eπ whenever lim Fα ({cn }, x) = F (x)

α→0

uniformly on compact subsets of R, where Fα ({cn }, x) =

∞ X

n=−∞

cn sinc(x − n)φα (x − n). 13

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Theorem 2 provides sufficient conditions P∞for this summability method to be regular. In other words, if f (x) = n=−∞ cn sinc(x − n) is a convergent cardinal sine series then Theorem 2 gives conditions that ensure that the corresponding F sum F (x) equals f (x). This is emphasized in the following Abelian type theorem stated as a corollary of Theorem 2. Corollary. The F summability method is regular when the family F = {φα (x)} satisfies Hypothesis H(ν) with ν ≥ 3. The converse of the above corollary is, in general, not true. Nevertheless, in the special case (29)

lim cn = 0

n→±∞

a converse result is indeed valid. Note that condition (29) by itself is not sufficient to ensure the convergence of the cardinal sine series with the coefficients {cn }. However if, in addition, the cardinal sine series is F summable then, in analogy with the classical Tauberian theorem [36, Subsection 1.23], so is the original cardinal sine series. More precisely:

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Theorem 3. Suppose the sequence {cn } satisfies (29) and the family F = {φα (x)} satisfies Hypothesis H(ν) with ν ≥ 1. Then, for every α in I, the series (6) converges absolutely and Fα ({cn }, x) is a well defined function in C ν (R). Furthermore, if for some real non-integer x = x0 lim Fα ({cn }, x0 )

α→0

then

exists and is finite

rep

(30)

f (z) =

∞ X

n=−∞

cn sinc(z − n)

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is a convergent cardinal sine series and lim Fα ({cn }, x) = f (x)

α→0

for all real x

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uniformly on compact subsets of R.

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Proof. Note that condition (29) implies that Theorem 1 applies with κ = 1. Hence if Fα ({cn }, x) converges at one non-integer x = x0 then it converges at all real x and

where

lim Fα ({cn }, x) = F (x) − c

α→0

sin(πx) π

∞ X cn F (x) = c0 sinc(x) + x sinc(x − n) n n=−∞ n6=0

14

Journal Pre-proof

is in Eπ and ∞ ∞ X X (−1)n bn (−1)n cn φα (n) = lim 2 φα (n). c = lim α→0 α→0 n n n=−∞ n=1 n6=0

where bn = (cn − c−n )/2. To establish that {cn } are the coefficients of a convergent cardinal sine series f (z) and that limα→0 Fα ({c}, x) = f (x) it suffices to show that ∞ X (−1)n bn

n

n=1

converges and equals

lim

α→0

∞ X (−1)n bn n=1

n

φα (n)

This follows from the next lemma which completes the proof of the theorem.

roo

f

Lemma 4. Suppose that bn , n = 1, 2, . . . , is a sequence such that lim bn = 0 and the family F = {φα (x)} satisfies hypothesis H(ν) with ν ≥ 1. n→∞ Then ∞ N X X bn bn lim = lim φα (n). α→0 N →0 n n n=1 n=1 In other words, the existence and finiteness of one side of the above equality implies the existence and finiteness of the other side.

rep

Proof. The conclusion of the lemma is a consequence of the inequality ∞ ) ( N N X b X X b n n |bn | + (αN )ν max |bn | (31) φα (n) − ≤C α n≥N +1 n n n=1 n=1 n=1

rna

To see (31) write ∞ X bn n=1

n

φα (n) −

N X bn n=1

n

=

N X bn n=1

n

{φα (n) − 1} +

where

∞ X bn φα (n). n n=N +1

  N N X b |φα (n) − 1| X n {φα (n) − 1} ≤ max |bn | 1≤n≤N n=1 n n n=1

Jo u

Since

lP

where C is a constant independent of α and N . This essentially follows from the fact that the left hand side of (31) can be made arbitrarily small by taking 1 ≤ αN ≤ 2 and choosing N or 1/α sufficiently large.

Z |φα (n) − 1| 1 n 0 = φα (x)dx ≤ max |φ0α (x)| ≤ Cα 0≤x≤n n n 0 15

Journal Pre-proof

and

where

( ∞ ∞ ) X b X |φα (n)| n φα (n) ≤ max |bn | n≥N +1 n n n=N +1 n=N +1 ∞ ∞ X X |φα (n)| 1 C C ≤ ≤ , ν n n |αn| |αN |ν n=N +1 n=N +1

inequality (31) follows.

Finally, to see how the right hand side of (31) can be made arbitrarily small by taking 1 ≤ αN ≤ 2 and choosing N or 1/α sufficiently large, let  > 0 and let N be such that both N 1 X |bn | <  and N n=1

max |bn | < 

n≥N +1

whenever N or 1/α is > N . Inequality (31) now implies that with this choice of N or α the left hand side of (31) is ≤ C.

Piecewise polynomial cardinal splines

3.1

Basic setup

rep

3

roo

f

This completes the proof of the lemma.

lP

If k is a positive integer, the space Sk of cardinal piecewise polynomial splines of order 2k with knots at the integers can be defined as consisting of functions s(x) that satisfy the following properties: (i) In each interval n ≤ x ≤ n + 1, n = 0, ±1, ±2, . . . , s(x) is a polynomial of degree no greater than 2k − 1.

rna

(ii) s(x) is in C 2k−2 (R)

Jo u

(iii) s(x) has no greater than polynomial growth as x → ±∞. In other words, |s(x)| ≤ C(1 + |x|)m for some constants C and m that are independent of x. The space Sk can also be described, more succinctly, as the class of tempered distributions s that satisfy s

(2k)

(x) =

∞ X

n=−∞

16

cn δ(x − n)

Journal Pre-proof

for some numerical sequence {cn }, where δ(x) is the Dirac measure at the origin, [20, Proposition 2]. Members of Sk are uniquely determined by their values on the integers, {s(n) : n = 0, ±1, ±2, . . .}. Conversely, for every bi-infinite sequence {cn : n = 0, ±1, ±2, . . .} of no greater than polynomial growth there is a unique spline s(x) in Sk that interpolates {cn }, namely s(n) = cn for n = 0, ±1, ±2, . . . . These facts are established in [33] and [20, Theorem1]. To indicate the dependence of this spline on k and on the sequence {cn } we denote it by Sk ({cn }, x). The function Lk (x) is the sampling kernel in Sk , the fundamental spline of order 2k that interpolates the Kronecker delta sequence {δ0,n }, namely ( 1 when n = 0 Lk (n) = δ0,n = 0 when n = ±1, ±2, . . . . Our development makes use of the Fourier transform of Lk (x), [20, (17) and Proposition7], ξ −2k . −2k j=−∞ (ξ + 2πj)

roo

f

bk (ξ) = P∞ L

(32)

rep

In the normalization used here, the Fourier transform fb(ξ) of an integrable function f (x) is defined as Z ∞ fb(ξ) = e−iξx f (x)dx −∞

and distributionally otherwise.

lP

Several more or less well known properties of Lk (x) are described in Lemmas 5 and 6.

rna

Lemma 5. There are positive constants, A and a, independent of x such that |Lk (x)| ≤ Ae−a|x| and |L0k (x)| ≤ Ae−a|x| .

Jo u

bk (ζ) is holomorphic in a strip Proof. This is a consequence of the fact that L {ζ = ξ + iη : |η| < } for some positive  and integrable on lines parallel to the real axis. Namely, if 0 < a <  then Z ∞ ±ax ±ax bk (ξ)eiξx dξ 2πe Lk (x) = e L −∞ Z ∞ Z ∞ i(ξ∓ia)x bk (ξ)e bk (ξ ± ia)eiξx dξ = L dξ = L −∞

−∞

17

Journal Pre-proof

so

±ax e Lk (x) ≤ 1 2π

Z

∞

−∞

bk (ξ ± ia)|dξ L

and hence the desired bound on |Lk (x)| follows.

Analogous reasoning, mutatis mutandis, justifies the bound on |L0k (x)|. Lemma 6. lim Lk (x) =

k→∞

sin πx πx

uniformly on R.

Proof. Note that (33)

bk (ξ) ≤ ξ −2k (ξ − 2πn)2k 0≤L

or, more crudely,

when |(ξ − 2πn)| ≤ π.

bk (ξ) ≤ ξ −2k π 2k . 0≤L

which is valid for all ξ. The last inequality together with (33) in the case n = 1 imply that for all ξ

Now

bk (ξ) = L

1+

1

P

|j|≥1

(ξ/(ξ − 2πj))2k

rep

from which it follows that

roo

f

bk (ξ) ≤ min{1, (ξ/π)−2k }. 0≤L

(34)

lP

  if |ξ| < π 1 b lim Lk (ξ) = 1/2 if |ξ| = π k→∞   0 if |ξ| > π.

In view of (34) and the dominated convergence theorem we may conclude that the last limit is valid in L1 (R) which implies the desired conclusion.

Jo u

(35)

rna

The spline Sk ({cn }, x) exists, is unique, and enjoys the representation Sk ({cn }, x) =

∞ X

n=−∞

cn Lk (x − n).

See [33] or [20, Theorem 2]. In view of Lemma 5 the symmetric partial sums of the series (35) converge absolutely and uniformly on compact subsets of R for every data sequence of samples {cn } of no greater than polynomial growth. 18

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Our development is based of the observation that the sampling kernel Lk (x) can be expressed as a damped cardinal sine. To begin, let (36)

Qk (x) =

Lk (x) πx Lk (x) = sinc(x) sin πx

and note that, given any integer n, (36) may be re-expressed as (37)

Qk (x) = x

Lk (x) − Lk (n) π(x − n) . n (−1) sin π(x − n) x−n

Together, these expressions imply   1   n 0 Qk (x) = (−1) nL (n)    πx Lk (x) sin πx

that when k > 1 when x = 0, when x = n, n = ±1, ±2, . . . , otherwise.

roo

f

In view of these identities, for every k, k = 1, 2, . . . , the function Qk (x) is a well defined function of the real variable x that is analytic between the integers and, if k > 1, continuous at the integers. In the case k = 1, because L1 (x) = (1 − |x|)+ , Qk (x) has simple jump discontinuities at x = ±1. In fact,

Qk (x) ∈ C 2k−3 (R) when k ≥ 2.

rep

(38)

To see this, note that in a neighborhood of any integer n

lP

c(x − n)2k−1 + (x − n)P (x) Lk (x) + = n sin πx (−1) π(x − n){1 − g(x)} = {c(x − n)2k−2 + P (x)} +

(

)
2 1 + g(x) + g(x) + · · · (−1)n π

That

Jo u

(39)

rna

where c is a constant, P (x) is a polynomial
of degree no greater than 2k − 2, and g is analytic with g(x) = O (x − n)2 as x − n → 0. Qk (−x) = Qk (x)

and

(40)

Qk (0) = 1

follow from the definition (36) and the corresponding properties of Lk (x). 19

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Lemma 7. There are positive constants, A and a, independent of x such that |Qk (x)| ≤ Ae−a|x| . Proof. Choose n such that |x − n| ≤ 1/2 then, in view of (37), write |Qk (x)| ≤

π 2

|x| |L0k (xn )| where xn is a point between x and n.

Now, if a is positive e−a|xn | ≤ e−a(|x|−|x−xn |) ≤ ea/2 e−a|x| and we may conclude that the estimate on |L0k (x)| in Lemma 5 implies the desired bound on |Qk (x)|.

3.2

Technical lemmas

roo

f

The desired result concerning the limiting behavior of Sk ({cn }, x) when k tends to ∞ will follow if the family {Qk (x) : k = 2, 3, . . .} satisfies an appropriate analog of the properties described by Hypothesis H(ν). This is established via a sequence of technical lemmas.

rep

Lemma 8. The Fourier transform of Qk (x) enjoys the representation bk (ξ) = −2π Q

(41)

∞ X

m=0

b0 (ξ + (2m + 1)π). L k

lP

Proof. Note that (36) can also be expressed as {sin πx}Qk (x) = πxLk (x)

(42)

rna

which implies that

bk (ξ + π) − Q bk (ξ − π) = 2π L b0k (ξ). Q

From (42) it follows that

Jo u

n o b0k (ξ + 2π) + · · · L b0k (ξ + 2πn) bk (ξ + π(2n + 1)) − Q bk (ξ − π) = 2π L b0k (ξ) + L Q

and since, in view of Lemma 7 and the Riemann-Lebesgue lemma, bk (ξ + π(2n + 1)) = 0 lim Q

n→∞

20

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we may conclude that bk (ξ − π) = 2π −Q

∞ X

m=0

b0k (ξ + 2πm). L

The desired result follows by replacing ξ in the last expression with ξ + π and multiplying both sides by -1. bk (ξ) can also be expressed as Note that Q bk (ξ) = 2π Q

(43)

∞ X

m=0

This follows from the fact that ∞ X

m=−∞

b0 (ξ − (2m + 1)π) . L k

bk (ξ + 2πm) = 1 which implies that L

∞ X

m=−∞

b0 (ξ + 2πm) = 0. L k

roo

f

In view of (41), it should be possible to obtain bounds on the derivatives of bk (ξ). Indeed, as we Qk (x) in terms of appropriate estimates of derivatives of L bk (ξ) found in the statement shall show, the estimates of the derivatives of L of Lemma 11 are sufficient to obtain the desired bounds on the derivatives of Qk (x). But first, it is necessary to establish two technical lemmas.

rep

To this end it is convenient to use the notation P =

∞ X

j=−∞

(ξ − 2πj)−2k

where the dependence of P on ξ and k is is not explicitly expressed.

rna

lP

Lemma 9. Suppose n is any integer. If |ξ − 2πn| ≤ π and 2k ≥ ν then using the notation established above (ν) P ν −ν (44) P ≤ C k |ξ − 2πn|

Jo u

where C is a constant independent of ξ and k and −2k−ν ξ −2k (45) |ξ − 2πn|2k−ν . P ≤ξ

Proof. To see (44) write ( ) P −2k−ν (ξ − 2πj) P (ν) (ξ − 2πn)−2k−ν j6=n = A(ν, k) + , P P P 21

Journal Pre-proof

where A(ν, k) = (−1)ν (2k)(2k + 1) · · · (2k + ν − 1). Observe that when |ξ − 2πn| ≤ π then 1/P ≤ (ξ − 2πn)2k ≤ π 2k , and hence ) ( (ν) 2k X P 1 π , (46) P ≤ |A(k, ν)| |ξ − 2πn|ν + |ξ − 2πj|2k+ν j6=n where the sum is taken of all integers j that are not equal to n. Since

|ξ − 2πj| = |ξ − 2πn − 2π(j − n)| ≥ |2π(j − n)| − |ξ − 2πn| ≥ π(|2(j − n)| − 1) it follows that (47)

X j6=n

∞

X π 2k 2 ≤ π −ν ≤ 4|ξ − 2πn|−ν , 2k+ν 2k+ν |ξ − 2πj| (2j − 1) j=1

where we used the fact that Z

0

∞

dt =2 (2t − 1)2

f

j=1

∞

X 1 1 ≤ ≤1+ 2k+ν 2 (2j − 1) (2j − 1) j=1

and π −ν ≤ |ξ − 2πn|−ν .

roo

∞ X

Since |A(ν, k)| < 3ν k ν , (46) and (47) imply the desired result. To see (45) proceed as follows:

rep

If |ξ| < π then

−2k−ν ξ −2k−ν 2k ξ = ξ −2k |ξ|2k−ν P ≤ |ξ|

which is (45) in the case n = 0.

rna

lP

In the case n 6= 0 write −2k−ν ξ −2k−ν |ξ − 2πn|2k = |ξ|−2k−ν π 2k |(ξ/π) − 2n|2k P ≤ |ξ| ≤ |ξ|−2k−ν π 2k |(ξ/π) − 2n|2k−ν = |ξ|−2k−ν π ν |ξ − 2πn|2k−ν = |ξ|−2k |ξ/π|−ν |ξ − 2πn|2k−ν ≤ ξ −2k |ξ − 2πn|2k−ν

Jo u

where we used the fact that |(ξ/π) − 2n| ≤ 1 and |ξ/π| ≥ 1. This establishes (45) and concludes the proof of the lemma. Lemma 10. Suppose n is any integer, |ξ − 2πn| ≤ π, and 2k ≥ ν. Then there is a constant C that is independent of k and ξ such that (48)

b (ξ)| ≤ Ck ν ξ −2k |ξ − 2πn|2k−ν . |L k (ν)

22

Journal Pre-proof

Proof. We argue by induction on ν. bk (ξ) and let P and A(ν, k) be as in Lemma 9. For convenience let L = L

For ν = 0

|L(0) | = |L| ≤ ξ −2k |ξ − 2πn|2k

which verifies (48) in this case. To see (48) for the wider case of ν > 0, assume true for 0, 1, . . . , ν − 1. To get an expression for L(ν) in terms of lower order derivatives of L, use the fact that LP = ξ −2k and differentiate both sides ν times to get ν   X ν (ν−µ) (µ) L P = A(ν, k) ξ −2k−ν . µ µ=0 Hence

f

L

ν   ξ −2k−ν X ν (ν−µ) P (µ) = A(ν, k) − L . µ P P µ=1

roo

(49)

(ν)

In view of the induction assumption and Lemma 9, |P (µ) | ≤ Ck ν−µ ξ −2k |ξ−2πn|2k−ν+µ k µ |ξ−2πn|−µ = Ck ν ξ −2k |ξ−2πn|2k−ν |P |

rep

|L(ν−µ) | and

−2k−ν ξ ≤ Ck ν ξ −2k |ξ − 2πn|2k−ν . |A(ν, k)| P

lP

This together with (49) imply the desired result.

rna

In the statement of Lemma 10, since ν ≤ 2k the term |ξ − 2πn|2k−ν is dominated by π 2k−ν = π −ν π 2k which is less than π 2k . Hence |ξ − 2πn|2k−ν ≤ π 2k . Using this last inequality in (48) results in an inequality that is independent of n and hence valid for all ξ. We formally record this fact as follows:

Jo u

Lemma 11. If 2k ≥ ν then there is a constant C that is independent of k and ξ such that (50)

for all real ξ.

b (ξ)| ≤ Ck ν (ξ/π)−2k |L k (ν)

b(ν) (ξ). The last lemma gives rise to the following estimate of Q k 23

Journal Pre-proof

bk (ξ) of order ν satisfies Lemma 12. If 2k ≥ ν + 1 then the derivative of Q b (ξ)| ≤ C k ν+1 (1 + |ξ/π|)−2k+1 |Q k (ν)

(51)

where the constant C is independent of k and ξ. Proof. Recall that bk (ξ) = Q b(0) (ξ) = −2π Q k

and hence

b(ν) (ξ) = −2π Q k

(52)

∞ X j=0

In view of Lemma 11,

b |L k

Hence ∞ X j=0

b |L k

(ν+1)

b(ν+1) (ξ + (2j + 1)π). L k

(ξ)| ≤ C k ν+1 (ξ/π)−2k .

(ξ + (2j + 1)π)| ≤ C k ν+1

Now, if η ≥ 1 then

∞

(ξ/π + 1 + 2j)−2k .

j=0

(η + 2t)−2k dt = η −2k +

rep

Z ∞ X −2k −2k (η + 2j) ≤η +

0

j=0

∞ X

roo

(53)

j=0

b0 (ξ + (2j + 1)π) L k

f

(ν+1)

∞ X

η −2k+1 ≤ 2η −2k+1 . 2(2k − 1)

Utilizing this with (53) results in

j=0

b |L k

(ν+1)

(ξ + (2j + 1)π)| ≤ C k ν+1 (1 + ξ/π)−2k+1 .

lP

∞ X

rna

b(ν) (−ξ)| = |Q b(ν) (ξ)| the last displayed inequality and when ξ ≥ 0. Since |Q k k (52) imply the desired result and completes the proof.

Jo u

Before moving on to the next lemma, we note that for non-negative integers j and m Z ∞ j!(m − j − 2)! (54) ξ j (1 + ξ)−m dξ = if 0 ≤ j ≤ m − 2. (m − 1)! 0 This will be useful in what follows. Identity (54) is evident when j = 0. When j > 0 integration by parts and induction yield the desired result. 24

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Lemma 13. If 2k − 3 ≥ j then for every non-negative integer ν there is a constant C independent of k and x such that (j)

|Qk (x)| ≤

(55)

Ck −j . 1 + |x/k|ν

Proof. Note that ν j b 1 d {ξ Qk (ξ)} (j) |xν Qk (x)| ≤ dξ. 2π −∞ dξ ν Z

∞

To estimate the right hand side of the above inequality write

∞

−∞

|ξ

j−`

Next, by virtue of (51),

b(ν−`) (ξ)|dξ ≤ Ck ν−`+1 Q k

Z

∞

ξ j−` (1 + ξ/π)−2k+1 dξ

f

Z




ν j! . ` (j−`)!

and, by virtue of (54), Z ∞ ξ j−` (1 + ξ/π)−2k+1 dξ =

π j−`+1 (j − `)! C ≤ j−`+1 . (2k − 2) · · · (2k − (j − `) − 2) k

rep

0

0

roo

where c` =

X bk (ξ)} min{j,ν} dν {ξ j Q b(ν−`) (ξ) = c` ξ j−` Q k dξ ν `=0

Combining the implications of the last four displayed expressions in the cases ν = 0 and ν = ν results in (j)

lP

|Qk (x)| ≤ Ck −j min{1, (x/k)−ν } which implies (55) and completes the proof.

The behavior of Sk ({cn }, x) as k tends to ∞.

rna

3.3

Jo u

In view of (36) the piecewise polynomial spline sampling kernel Lk (x) can be regarded as the damped cardinal sine Lk (x) = sinc(x)Qk (x)

so that Sk ({cn }, x) can be expressed as (56)

Sk ({cn }, x) =

∞ X

n=−∞

cn sinc(x − n)Qk (x − n). 25

Journal Pre-proof

Using the identification α = 1/k and φα (x) = Qk (x), the development in Subsections 3.1 and 3.2 shows that the family of functions F = {φα (x) = Qk (x)} satisfies hypothesis H(ν) when 2k−3 ≥ ν. We formulate this formally as follows: Lemma 14. For every k = 2, 3, . . . , the function Qk (x) is continuous and (57)

Qk (0) = 1

and

Qk (−x) = Qk (x).

Furthermore, for every positive integer ν the family of functions F = {φα (x) = Qk (x) : k ≥ k0 } satisfies hypothesis H(ν) when 2k0 − 3 ≥ ν. In other words, if 2k − 3 ≥ ν then Qk (x) is in C ν (R) and (j)

|Qk (x)| ≤

(58)

Ck −j , 1 + |x/k|ν

j = 0, 1, . . . , ν,

where C is a constant independent of k and x.

roo

f

In view of Lemma 14 the results of Section 2 lead to several conclusions concerning the behavior of Sk ({cn }, x) as k tends to ∞. As a corollary of Theorem 2 we have the following. Theorem 4. The cardinal spline summability method is regular. That is, if cn sinc(z − n)

rep

f (z) =

∞ X

n=−∞

is a convergent cardinal sine series then lim Sk ({f (n)}, x) = f (x)

lP

k→∞

uniformly on compact subsets of R.

Jo u

rna

Since Sk ({P (n)}, x) = P (x) whenever P (x) is a polynomial of degree no greater than 2k − 1, it follows that Theorem 4 cannot have an actual converse. However, as a corollary to Theorem 3, a Tauberian type result is valid. Namely, if in addition f (n) = o(1) as n tends to ±∞ then a converse result is valid. Viewed from a different perspective, the condition cn = o(1) as n tends to ±∞ is not sufficient to ensure that {cn } be the coefficients of a convergent cardinal sine series; the convergence of Sk ({cn }, x) provides an additional condition that, according to the next theorem, is sufficient to ensure the desired result. 26

Journal Pre-proof

Theorem 5. Suppose the sequence {cn } satisfies lim cn = 0.

n→±∞

If for some real non-integer x = x0 lim Sk ({cn }, x0 )

exists and is finite

k→∞

then f (z) =

∞ X

n=−∞

cn sinc(z − n)

is a convergent cardinal sine series and lim Sk ({cn }, x) = f (x)

k→∞

for all real x.

As a corollary of Theorem 1 we have

roo

f

Theorem 6. Suppose the data sequence {cn } has no greater than polynomial growth. Then as k tends to ∞ the splines Sk ({cn }, x) either (a) converge to a function F in Eπ uniformly on compact subsets of R or (b) diverge for all but at most a finite number of non-integer values of x. More specifically, if cn = O(nρ ) as n → ±∞ where 0 ≤ ρ < κ then in the case (a)

rep

∞ n o X cn sin(πx) F (x) = c0 sinc(x) + xκ sinc(x − n) − P (x) κ n π n=−∞ n6=0

lP

where P (x) is a polynomial of degree ≤ κ − 1 and in the case (b) the sequence {Sk ({cn }, x); k = 1, 2, 3, . . .} can converge at no more than κ − 1 non-integer values of x.

rna

Finally, for the sake of completeness, we mention that it is known that (5) is valid whenever the function f is in Eσ for σ < π, [21, 30]. Combining this with the result stated in Theorem 4 leads to the following conclusion.

Jo u

Proposition. Suppose f (z) = f1 (z) + f2 (z) where f1 (z) is a function in Eσ where σ < π and f2 (z) is a convergent cardinal sine series. Then lim Sk ({f (n)}, x) = f (x)

k→∞

uniformly on compact subsets of R.

27

Journal Pre-proof

3.4

Spline summability and the Bernstein class Bπ

The Bernstein class Bπ consists of those functions f (z) in Eπ that are bounded on the real axis. Because sin πx is in Bπ the values {f (n) : n = 0, ±1, ±2, . . .} do not uniquely determine a function f (z) in Bπ . Nevertheless, in [35] Schoenberg articulated the following conjecture: Conjecture 1. If f is in Bπ then for some constant c (59)

lim Sk ({f (n)}, x) = f (x) + c sin πx

k→∞

uniformly in x on compact subsets of R. This was partially motivated by (i) the fact that (59) was known to be valid for certain subclasses of functions f in Bπ and (ii) as a natural converse to the fact that if (59) is valid for a bounded continuous function f then f must be a member of Bπ , [35].

roo

f

In the case that f in Bπ is even, namely f (−z) = f (z), (59) is valid with the constant c = 0. This is true by virtue of the fact that such functions f are convergent cardinal sine series and Theorem 4. We formalize this as follows: Theorem 7. Suppose f is an even function in Bπ , namely f (−z) = f (z). Then lim Sk ({f (n)}, x) = f (x)

rep

k→∞

uniformly in x on compact subsets of R.

lP

Proof. The sequence of samples {f (n)} is bounded and even which implies that they are the samples of a convergent cardinal sine series g(z) that is even with growth no greater than O(log |x|) as |x| → ∞ on the real axis, [25]. It follows that f (z) − g(z) = c sin πz, [19, Theorem 1, p. 155]. Since the left hand side of the last identity is even, the constant c is 0.

2k

∞ 2X −1 sin πz X 1 2z 2 k F (z) = (−1) · π k=1 log n n(z 2 − n2 ) k−1

Jo u

(60)

rna

However Conjecture 1 is not necessarily valid in the general case. In view of Theorem 7, it suffices to look for possible counter examples among the odd functions in Bπ . To this end consider the odd function

n=22

which is studied in some detail in [5, Theorem 3], where it is shown that series (60) converges absolutely and uniformly on compact subsets of C to an entire function F (z) in the Bernstein class Bπ . (A related entire function 28

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with analogous properties is studied in [4].) The function F (z) may also be expressed as ∞ X 2z sin π(z − n) z · F (z) = cn · · n z+n π(z − n) n=1 or

∞ X cn sin π(z − n) F (z) = z · . n π(z − n) n=−∞

where cn = 0 if |n| = 0 or 1 and cn =

(−1)|n|+k n · log |n| |n|

k−1

if 22

k

≤ |n| ≤ 22 − 1, k = 1, 2, 3, . . . .

Note that F (z) is an odd function of z, namely F (−z) = F (z), and (−1)n+k log n

k−1

when 22

Since the series

k

≤ n ≤ 22 − 1 , k = 1, 2, . . . .

∞ X (−1)n F (n)

f

F (n) =

n

roo

n=1

fails to converge, F (z) is not a convergent cardinal sine series. In view of Theorem 5 we may conclude that the cardinal splines Sk ({F (n)}, x) also fail to converge as k → ∞. We summarize this as follows:

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Theorem 8. The function F (z) defined by (60) is a function in the Bernstein class Bπ such that the cardinal splines Sk ({F (n)}, x) fail to converge as k → ∞ for every real non-integer value of x.

Closing Remarks

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4

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Corollary. Conjecture 1 is false.

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4.1 Sampling series involving kernels consisting of damped cardinal sines have been studied by various authors, including [7, 24, 27, 28, 32]. Such sampling series are often referred to as regularizations of the classical cardinal sine series and are used as approximations. To my knowledge, the results in Section 2 are new. 4.2 In [5] we studied the case where the family F = {φα (z) : α ∈ I} is of the form φα (z) = ψ(αz), with 0 < α ≤ 1 and with ψ(z) a function in E1 that decays sufficiently rapidly along the real axis. It was shown that 29

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the corresponding summability method is regular and converges on compact subsets of the complex plane C. The methods employed here in Section 2 can be used to obtain summability results valid in the whole complex plane C in the case that the function ψ(z) is holomorphic in a horizontal strip {z : | Im z| < } and decays sufficiently rapidly as Re z tends to ±∞ in the strip. 4.3 In view of Theorems 2 and 4 the sampling series considered here give rise to regular summability methods for the classical cardinal sine series. Summability methods associated with the names of Cesaro and Abel have also been considered for the classical cardinal sine series by various authors, including [11, 37, 39]. It is known that such methods have good convergence properties when the sequence of data samples {cn } is sufficiently well behaved but, unlike the methods involving damped cardinal sines, members of the sequence of approximants do not, in general, interpolate the data.

rep

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f

4.4 In [29] it was shown that if f (x) is bounded on R and limk→∞ Sk ({f (n}, x) = f (x) uniformly on R then f is in Bπ . Subsequently Schoenberg observed in [35] that the result is still valid when, for some constant c, limk→∞ Sk ({f (n}, x) = f (x) + c sin πx uniformly on compact subsets of R. (Conjecture 1 mentioned in Subsection 3.2 was in part intended as a natural converse.) Theorem 6 represents a significant improvement of this observation. For instance, Theorem 6 implies the following:

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If the sequence {cn } is bounded and, as k tends to ∞, Sk ({cn |}, x) converges for one non-integer point x = x0 then Sk ({cn }, x) converges uniformly on compact subsets of R to a function F (x) in Eπ as k tends to infinity.

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4.5 The families of sampling kernels considered here are a subset of a general class of families whose limiting kernel is in some sense frequency band limited. Other examples of such families of sampling kernels can be found, for instance, in [1, 2, 6, 9, 10, 12, 13, 18, 22, 23, 31]. In most of these cases it is known that the corresponding sampling series converge to a frequency band limited function when the sequence of the data samples is sufficiently well behaved; usually in `2 which, in many cases, seems far more restrictive than necessary. It is quite likely that results analogous to Theorems 1- 6 are valid for a wider collection of families in this class. A general theory is beyond the scope of this article.

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[5] B. A. Bailey and W. R. Madych, Convergence and summability of cardinal sine series, Jaen J. Approx. 10, No. 1-2 (2018), 49-72.

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[6] C. de Boor, K. H¨ollig, S. Riemenschneider, Convergence of cardinal series. Proc. Amer. Math. Soc. 98 (1986), no. 3, 457-460.

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[7] P. L. Butzer and R. L. Stens, A modification of the WhittakerKotel’nikov-Shannon sampling series, Aequation’es Mathematicae 28 (1985), 305-311. [8] P. L. Butzer, J. R. Higgins, and R. L. Stens, Sampling theory of signal analysis, Development of mathematics 1950-2000, 193-234, Birkh¨auser, Basel, 2000.

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[9] M. Chang, The behavior of polyharmonic cardinal splines as their degree tends to infinity, J. Approx. Theory 76, (1994), no. 3, 287-302.

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[10] G. Deslauriers and S. Dubuc, Symmetric iterative interpolation processes. Constr. Approx. 5 (1989), no. 1, 49-68. [11] R. Estrada, Summability of cardinal series and of localized Fourier series, Appl. Anal. 59, no. 1-4 (1995), 271-288.

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[12] T. N. T. Goodman and S. L. Lee, A remainder formula and limits of cardinal spline interpolants. Trans. Amer. Math. Soc. 271 (1982), no. 2, 469-483. [13] K. Hamm and J. Ledford, Cardinal interpolation with general multiquadrics. Adv. Comput. Math., 42 (2016), no. 5, 1149-1186. 31

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[21] W. R. Madych, Polyharmonic splines, multiscale analysis and entire functions, Multivariate approximation and interpolation (Duisburg, 1989), 205-216, Internat. Ser. Numer. Math., 94, Birkh¨auser, Basel, 1990.

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[22] W. R. Madych, Miscellaneous error bounds for multiquadric and related interpolators. Advances in the theory and applications of radial basis functions. Comput. Math. Appl., 24 (1992), no. 12, 121-138. [23] W. R. Madych, Spline type summability for multivariate sampling, in Analysis of divergence (Orono, ME, 1997), 475-512, Appl. Numer. Harmon. Anal., Birkhuser Boston, Boston, MA, 1999.

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[24] W. R. Madych, Summability of Lagrange type interpolation series, J. Anal. Math. 84 (2001), 207-229.

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[25] W. R. Madych, Convergence of Classical Cardinal Series, in Multiscale Signal Analysis and Modeling, X. Shen and A. I Zayed (eds.) Lecture Notes in Electrical Engineering, Springer (2012), 3-24. [26] W. R. Madych, On the convergence of cardinal splines, to appear in ACHA.

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[33] I. J. Schoenberg, Cardinal spline interpolation, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. vi+125 pp.

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[34] I. J. Schoenberg, Cardinal interpolation and spline functions. VII. The behavior of cardinal spline interpolants as their degree tends to infinity. J. Analyse Math. 27 (1974), 205-229.

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[35] I. J. Schoenberg, On the remainders and the convergence of cardinal spline interpolation for almost periodic functions. Studies in spline functions and approximation theory, pp. 277-303. Academic Press, New York, 1976. [36] E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford University Press, London (1939).

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[37] G. G. Walter, Abel summability for a distribution sampling theorem, Generalized functions, convergence structures, and their applications (Dubrovnik, 1987), 349-357, Plenum, New York, 1988.

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[38] G. G. Walter and X. Shen, Wavelets and other orthogonal systems, 2nd ed. Studies in Advanced Mathematics. Chapman and Hall/CRC, Boca Raton, FL, 2001. [39] J. M. Whittaker, Interpolatory function theory, Cambridge Tracts in Mathematics and Mathematical Physics, No. 33, Cambridge University Press, Cambridge (1935). 33

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[40] A. I. Zayed, Advances in Shannon’s sampling theory. CRC Press, Boca Raton, FL, 1993.

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