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Approximation of entire functions of exponential type by trigonometric polynomials.

Abstract

Let f be an entire function of exponential type [sigma] bounded on the real line. We associate with f an explicitly given function [f.sub.L,0] which coincides with a trigonometric polynomial of period L and exponential type [sigma] + 2[pi]/L on the strip [absolute value of Rz] [less than or equal to] < L/2 and which is zero outside. When f(x) [right arrow] 0 as x [right arrow] [+ or -][infinity], we show that [f.sub.L,0] converges to f as L [right arrow] [infinity] uniformly on horizontal strips. We also prove convergence with respect to [L.sup.p] norms on R and on lines parallel to R and convergence with respect to [l.sup.p] norms. These results include a simple and more general alternative answer to a question which was recently raised and answered by R. Martin.

Key words and phrases : entire functions of exponential type, Bernstein spaces, approximation by trigonometric polynomials

2000 AMS Mathematics Subject Classification - 30D15, 42A10

1 Introduction

Let f be an entire function of exponential type [LAMBDA], where [LAMBDA] < [OMEGA], and of finite [L.sup.2] norm on R. In a recent paper, Martin [7] (1) has asked for a trigonometric polynomial [f.sub.L] of given period L and exponential type [OMEGA] such that [f.sub.L] approaches f as L [right arrow] [infinity]. Here we use the notion of exponential type as introduced in [2, p. 8] and prefer it to the term bandlimited.

In answering his question, Martin proceeded (in a somewhat different notation) as follows:

* Restrict f to the strip -L/2 < Rz [less than or equal to] L/2 and let [[??].sub.L] be the L-periodic continuation of this restriction.

* Expand [[??].sub.L](z) on [-L/2, L/2] in a Fourier series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

* Set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where N := [??][OMEGA]L/(2[pi])[??], to obtain a trigonometric polynomial [f.sub.L] of period L and exponential type [OMEGA].

Denoting by [[phi].sub.L] the function which coincides with [f.sub.L] on the strip [absolute value of Rz] [less than or equal to] L/2 and which is zero outside, Martin proved that [[phi].sub.n] converges to f in the [L.sup.2] norm on R and on lines parallel to R as L [right arrow] [infinity]. He also proved uniform convergence on horizontal strips and convergence in an [l.sup.2] norm.

Here we show that the same conclusions, for more general norms and partly under a weaker hypothesis, can be obtained easier by using an alternative trigonometric polynomial [f.sub.L] proposed by Hormander [5, p. 40] in 1954; also see [6].

As usual, let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Given L > 0, Hormander associates with f the function

[f.sub.L](z) := [summation over (n[member of]Z)][sinc.sup.2](z/L + n)f(z + nL). (1)

Obviously, if [[florin[.sub.L] exists, then it is L-periodic. In the following proposition, we summarize further properties of [f.sub.L]. They can be obtained as in [6, pp. 22-28] with slight modifications of the proofs if necessary.

Proposition A Let f be an entire function of exponential type [sigma], bounded on the real line. Then the series in (1) converges absolutely and uniformly on horizontal strips, and

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] uniformly on all compact subsets of C.

An interesting feature of this trigonometric polynomial [f.sub.L] is that we need not know its coefficients [c.sub.n]. Using the representation (1), we can easily see that certain properties of f are inherited by [f.sub.L] and that certain properties of [f.sub.L] are preserved as L [right arrow] [infinity]. In fact, [f.sub.L] has been successfully used for extending results on trigonometric polynomials to entire functions of exponential type; see, e.g., [6, 8].

While statement (iii) gives convergence on compact sets only, we can obtain global convergence by working with the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which corresponds to the aforementioned function [[phi].sub.L].

2 Statement of the Results

Proposition 2.1 Let f be an entire function of exponential type such that f(x) [right arrow] 0 as x [right arrow] [+ or -]=[infinity]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

uniformly on every strip [absolute value of [??]z] [less than or equal to] d.

In the subsequent results, we shall use the notion of Bernstein space. The Bernstein space [B.sup.p.sub.[sigma]] consists of all entire functions of exponential type [sigma] whose restriction to R belongs to [L.sup.P](R); see [4, Definition 6.5]. The norm in [L.sup.P](R) is denoted by

[parallel]f[[parallel].sub.p]:= [([[integral].sup.[infinity].sub.[-[infinity]][[absolute value of f(x)].sup.p] dx).sup.1/p] (1 [less than or equal to] p < [infinity]).

Proposition 2.2 For p [member of] [1, [infinity]) and [sigma] > 0, let f [member of] [B.sup.p.sub.[sigma]]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

uniformly for [absolute value of y] [less than or equal to] d.

Finally, we state a result which includes convergence with respect to [l.sup.P] norms. We say that a sequence [([x.sub.j]).sub.j[member of]Z] of real numbers is uniformly separated (by [delta]) if there exists a positive number [delta] such that [x.sub.j+1] - [x.sub.j] [greater than or equal to] [delta] for all j [member of] Z.

Proposition 2.3 For p [member of] [1, [infinity]) and [sigma] > 0, let f [member of] [B.sup.p.sub.[sigma]]. Let ([x.sub.j])j[member of] z be a uniformly separated sequence of real numbers. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3 Some Auxiliary Results

Lemma 3.1 For z [member of] C, we have

[summation of (n[member of]z] [sinc.sup.2](z/L + n) [equivalent to] 1 (2)

and

[LAMBDA](z) := [summation of (n[member of]z][absolute value of [sinc.sup.2]](z/L + n)][less than or equal to] [e.sup.2[pi][absolute value of [??]z]/L]. (3)

Proof The equivalence (2) is an immediate consequence of a well-known expansion of 1/[sin.sup.2]([pi]z) in partial fractions; see [1, p. 75, 4.3.92] or [3, p. 44, 1.422 (4)].

For a proof of (3), we consider the compact rectangular domain K with vertices at [+ or -] c [+ or -] id. Since A is continuous on K, it attains a maximum M, say, at some point z * [member of] K. Now let [[theta].sub.n [member of] R be chosen such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This is an entire function of exponential type 2[pi]/L. On the real line we have, as a consequence of (2),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, by a standard result on entire functions of exponential type [2, Theorem 6.2.4], we obtain

[absolute value of F(z)] [less than or equal to] [e.sup.e[pi][absolute value of [??]z]/L] (z [member of] C),

and so

M = [LAMBDA](z*) = F(z*) [less than or equal to] [e.sup.2[pi][absolute value of [??][z*]/L].

The bound on the right-hand side does not depend on c. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies (3).

Next, we recollect some facts about entire functions of exponential type.

Lemma 3.2 Let f be an entire function of exponential type such that f(x) [right arrow] 0 as x [right arrow] [+ or -]. Then f(x + iy) [right arrow] 0 as x [right arrow] [+ or -] uniformly for y [member of] [-d, d].

Proof Since f is bounded on the real line, it is also bounded on horizontal strips [2, Theorem 6.2.4]. Therefore, the conclusion of the lemma follows by a variant of a result by Montel; see [2, Theorem 1.4.9].

For c [greater than or equal to] 0, y [member of] R, and p [greater than or equal to] 1, define

[F.sub.p](c, y) := [[integral].sub.[absolute value of x][greater than or equal to]c][[absolute value of f(x + iy)].sup.p] dx.

We now state a few properties of the function [F.sub.p].

Lemma 3.3 For p [member of] [1, [infinity]) and [sigma] > 0, let f [member of] [B.sup.p.sub.[sigma]]. Then [F.sub.p](c, y) exists and

(i) [F.sub.p](0, y) [less than or equal to] [e.sup.p[sigma][absolute value of y] [F.sub.P](0,0) = [e.sup.p[sigma][absolute value of y]][[parallel]f[[parallel].sup.p.sub.p],

(ii) [F.sub.p](c, *) is continuous,

(iii) [F.sub.p](c, y) [right arrow] 0 as c [right arrow] [infinity] uniformly for y [member of] [-d, d].

Proof Statement (i) is a standard result on functions f [member of] [B.sup.P.sub.[sigma]]; see [2, Theorem 6.7.1]. Its proof yields the existence of [F.sub.p](0,y) as well. The existence of [F.sub.p](c, y) for c > 0 is an obvious consequence.

For real h, it follows from (i) that

([e.sup.-p[sigma][absolute value of h] -1) [F.sub.p](0,y) [less than or equal to] [F.sub.p](0,y + h) - [F.sub.p](0, y) [less than or equal to] ([e.sup.p[sigma][absolute value of h] - 1)[F.sub.p](0,y).

This shows that [F.sub.p](0, *) is continuous. Since

[F.sub.p](c, y) = [F.sub.p](0, y) - [[integral].sup.c.sub.-c] [[absolute value of f(x + iy)].sup.p] dx,

we easily conclude that [F.sub.p](c,*) is also continuous.

Statement (iii) follows from (ii) by a compactness argument. In fact, let [epsilon] > 0 be given. For each y [member of] [-d, d], there exists a [c.sub.y] > 0 such that [F.sub.p]([c.sub.y], y) < [epsilon]/2. Because of (ii), there exists an open neighborhood [U.sub.y] of y such that [F.sub.p]([c.sub.y], [eta]) < [epsilon] for [eta] [member of] [U.sub.y]. Since [-d, d] is compact, it is covered by a finite number of neighborhoods [U.sub.y], say [-d, d] [subset] [U.sub.y1] [union] ... [union] [U.sub.yk]. Then, setting

c := max{[c.sub.y1], ..., [c.sub.yk]},

we have [F.sub.p](c, y) < [epsilon] for all y [member of] [-d, d]. This completes the proof.

4 Proofs of the Propositions

Proof of Proposition 2.1 The idea is to split [S.sub.d] := {z [member of] C : [absolute value of [??]z] [less than or equal to] d} into three disjoint parts:

[A.sub.1] := {z [member of] [S.sub.d] : [absolute value of Rz] [less than or equal to] c},

[A.sub.2] := {z [member of] [S.sub.d] : c < [absolute value of Rz] < L/2},

[A.sub.3] := {z [member of] [S.sub.d] : [absolute value of Rz] [greater than or equal to]L/2},

for appropriately chosen c and L.

Given [epsilon] > 0, there exists, by Lemma 3.2, a c > 0 such that

[absolute value of f(z)] < [epsilon] for z [member of] [S.sub.d] \ [A.sub.1. (4)

Now, by statement (iii) of Proposition A, there exists an [L.sub.0] > max{2c, 1} such that

[absolute value of f(z)] - [f.sub.L,0](z)] < [epsilon] for z [member of] [A.sub.1.

and L [greater than or equal to] [L.sub.0].

Next, let z [member of] [A.sub.2]. Then, using Lemma 3,1, we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we note that the arguments of f on the right-hand side always belong to [S.sub.d] \ [A.sub.1]. Therefore, (3) and (4) allow us to conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and L [greater than or equal to] [L.sub.0].

Finally, as a consequence of (4),

[absolute value of f(z) - [f.sub.L,0](Z)] = [absolute value of f(z)] < [epsilon] for z [member of] [A.sub.3].

This completes the proof.

Proof of Proposition 2.2 The proof is similar to that of Proposition 2.1. For appropriately chosen c > 0 and L > 2c, we split R into three disjoint parts:

[B.sub.1] := [-c, c], [B.sub.2] := (-L/2, -c) [union] (c, L/2), [B.sub.3] := R \ (-L/2, L/2).

Given [epsilon] > 0, there exists, by statement (iii) of Lemma 3.3, a c > 0 such that [F.sub.p](c, y) < [epsilon] for y [member of] [-d, d].

By statement (iii) of Proposition A, there exists an [L.sub.0] > max{2c, 1} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for y [member of] [-d, d] and L [greater than or equal to] [L.sub.0].

Next, we write

[[lambda].sub.n](z) := [sinc.sup.2](z/L + n)

for short, where z = x + iy, x [member of] [B.sub.2] and y [member of] [-d, d]. Noting that t [??] [[absolute value of].sup.p] is a monotonically increasing convex function on [0, [infinity]), we conclude, using Lemma 3.1, that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, let

[I.sub.n] := {x + nL : x [membre of] [B.sub.2]} (n [member of] Z).

Then [I.sub.n] [intersection] [I.sub.m] = 0 for n [not equal to] m and [[union].sub.n[subset]Z][I.sub.n] [subset] R \ [-c, c]. Using Levi's theorem for justifying an interchange of integration and summation, we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for L [greater than or equal to] [L.sub.0] and y [member of] [-d, d]. Finally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for L [greater than or equal to] [L.sub.0] and y [member of] [-d, d].

Altogether, we have

[parallel]f(* + iy) - [f.sub.L,0](* + iy)[[parallel].sub.p]. < [(2[epsilon] + [2.sup.p-1] [e.sup.p2[pi]d][epsilon]).sup.1/p]

for L [greater than or equal to] [L.sub.0] and y [member of] [-d, d]. This shows the uniform convergence.

Proof of Proposition 2.3 Suppose that the sequence [([x.sub.j]).sub.j[member of]Z] is uniformly separated by [delta] > 0. For any sequence [([[xi].sub.j])j[member of]Z] that is uniformly separated by [delta], it follows from [2, p. 101, lines 7-9] with [phi](t) = [[absolute value of t].sup.p], [delta] replaced by [delta]/2 and c > [delta]/2 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This allows us to proceed as in the previous proof. Let [B.sub.1], [B.sub.2], and [B.sub.3] be as before. Given [epsilon] > 0, there exists a c > [delta]/2 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Because of statement (iii) of Proposition A, there exists an [L.sub.0] > 2c such that

[summation over [x.sub.j][member of][B.sub.1]] [[absolute value of f(x.sub.j) - [f.sub.L,0]([x.sub.j])].sup.p] < [epsilon]

for L [greater than or equal to] [L.sub.0].

Next, let [x.sub.j] [member of] [B.sub.2]. Using the monotonicity and the convexity of the function t [??] [[absolute value of].sup.p] as in the previous proof, we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is easily seen that

X := {[x.sub.j] + nL : [x.sub.j] [member of] [B.sub.2], n [member of] Z} [subset] R\[-c,c].

Unfortunately, in the set X, two elements that are closest to an odd multiple of L/2 may have a distance less than [delta]. However, a short reflection yields that the elements of X can be arranged in two sequences [([s.sub.v]).sub.v[member of]Z] and [([t.sub.v]).sub.v[member of]Z], say, which are both uniformly separated by [delta]. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where (5) has been used in the last step.

Finally, again by (5),

[summation over ([x.sub.j][member of][B.sub.3])] [[absolute value of f([x.sub.j]) - [f.sub.L,0]([x.sub.j])].sup.p] = [[summation over ([x.sub.j][member of][B.sub.3])] [[absolute value of f([x.sub.j])].sup.p] < [epsilon],

Altogether, we have

[summation over (j[member of]Z)] [[absolute value of f([x.sub.j]) - [f.sub.L,0]([x.sub.j])].sup.p] = [less than or equal to] (2 + [2.sup.p])[epsilon]

for L [greater than or equal to] [L.sub.0]. This completes the proof.

References

[1] M. Abramowitz and I. A. Stegun (Ed.), Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, D.C., 1964.

[2] R.P. Boas, Entire Functions, Academic Press, New York, 1954.

[3] I.S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., Academic Press, Boston, 1994.

[4] J.R. Higgins, Sampling Theory in Fourier and Signal Analysis, Clarendon Press, Oxford, 1996.

[5] L. Hormander, A new proof and a generalization of an inequality of Bohr, Math. Scand. 2, 33-45, 1954.

[6] L. Hormander, Some inequalities for functions of exponential type, Math. Scand. 3, 21-27, 1955.

[7] R. Martin, Approximation of [ohm]-bandlimited functions by [OMEGA]-bandlimited trigonometric polynomials, Sampl. Theory Signal Image Process., 6,273-296, 2007.

[8] Q.I. Rahman and G. Schmeisser, [L.sup.p] inequalities for entire functions of exponential type, Trans. Amer. Math. Soc. 320, 91-103, 1990.

Gerhard Schmeisser

Mathematical Institute, University of Erlangen-Nuremberg, Bismarckstrasse 1 1/2 D-91054 Erlangen, Germany schmeisser@mi.uni-erlangen.de

(1) Editor note: The present paper is much related to the previous paper by Martin on pp. 273-296 of this issue
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Date:Sep 1, 2007
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