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Approximation of [OMEGA]-bandlimited functions by [OMEGA]-bandlimited trigonometric polynomials.

Abstract

It is known that the space of [OMEGA]-bandlimited functions is dense in [L.sup.2] norm on any finite interval [a, b]. In particular, for any [OMEGA] > 0 there exist so-called superoscillating [OMEGA]-bandlimited functions which can oscillate arbitrarily quickly on any finite interval of arbitrary length. This raises the question, is any [OMEGA]-bandlimited function in some sense the limit of a sequence of [OMEGA]-bandlimited trigonometric polynomials whose periods become infinite in length? Although the existence of superoscillating bandlimited functions may appear to suggest that the answer is negative, it is shown in this paper that any [OMEGA]-bandlimited function can indeed be seen both as the uniform pointwise limit on any compact set. and the [L.sup.2] limit on any line parallel to R of a sequence of spatially-truncated [OMEGA]-bandlimited trigonometric polynomials whose periods become infinite in length. That these results are indeed consistent with and supported by known results about superoscillations and [OMEGA]-bandlimited functions is explained.

Key words and phrases : trigonometric polynomials, bandlimited, Paley-Wiener space, reconstruction, sampling.

2000 AMS Mathematics Subject Classification--42A10,42A15,42A65.

1 Introduction

For any fixed strictly [OMEGA]-bandlimited function f this paper provides a method of constructing a sequence of [OMEGA]-bandlimited trigonometric polynomials {[f.sub.N}N[member of]N] which converge to the bandlimited function f uniformly on any compact set in C. It will be further shown that the members of this sequence of trigonometric polynomials {[f.sub.N]} can be multiplied by characteristic functions [chi]N of certain vertical strips of increasing width to yield a new sequence {[[phi].sub.N] := [f.sub.N[chi]N]} which converges to f in [L.sup.2] norm on rely line parallel to the real axis in the complex plane C. For any fixed [OMEGA]-bandlimited function, sequences of [OMEGA]-bandlimited trigonometric polynomials which converge to it have been constructed in the past [5, 15]. The sequences considered in this paper, however, can be seen as a more natural approximation of the original [OMEGA]-bandlimited function since they are directly the image of the original function under a sequence of spectral projections of self-adjoint Laplacians on spatial intervals of increasing size.

The above convergence results may appear to be in conflict with the fact that the space of [OMEGA]-bandlimited functions with an arbitrarily small but finite bandlimit [OMEGA] > 0 is dense in [L.sup.2] norm on any finite interval of arbitrarily large size [16]. For example, given any finite interval [a, b], and any positive value [OMEGA] > 0 one can construct a sequence of 'spheroidal prolate wave functions' which are [OMEGA]-bandlimited, form an orthonormal basis for the Hilbert space of [OMEGA]-bandlimited functions, B([OMEGA]), and which are simultaneously a complete orthogonal set in [L.sup.2][a, b] [16]. This means that, given any finite interval, one can draw a continuous function that oscillates arbitrarily quickly, and then find a sequence of [OMEGA]-bandlimited functions that converge to it in [L.sup.2] norm on that interval. Furthermore, one can specify any values at any finite (but arbitrarily large) number of points and find a bandlimited function which achieves those values at those points (1) [7, 6] . By choosing alternating positive and negative values on a densely packed finite set of points, one can thus construct a bandlimited function which displays 'superoscillatory' behaviour. That is, the resulting bandlimited function undergoes rapid changes on length scales much smaller than the shortest wavelength corresponding to the highest frequency component in its Fourier spectrum.

The phenomenon of superoscillations follows from long-known results of [10] [16, 11]. More recently superoscillations have been rediscovered in the field of mathematical physics [3], and are currently a subject of some interest in both the mathematical physics and sampling theory communities; see, e.g., [7, 6, 8, 2].

The existence of superoscillations shows that in some sense the space of [OMEGA]-bandlimited functions has an arbitrary number of 'degrees of freedom' in any finite interval [-L, L]. On the other hand, the set of all [OMEGA]-bandlimited trigonometric polynomials of period 2L with the [L.sup.2] inner product on [-L, L] forms a finite 2 [??][OMEGA]L/[pi][??] + 1 dimensional Hilbert space, [B.sub.L]([OMEGA]), so that any [OMEGA]-bandlimited trigonometric polynomial of period 2L has only a finite number of 'degrees of freedom' in the interval [-L, L]. Moreover, since [OMEGA]-bandlimited functions are [L.sup.2] dense on [-L, L], there exists a sequence {[f.sub.n]} of [OMEGA]-bandlimited functions which converges in [L.sup.2] norm to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on [-L, L], [k.sub.l] := l[pi]/L where [absolute value of l] can be chosen large enough so that [absolute value [k.sub.l]] > [OMEGA]. Since all [OMEGA]-bandlimited trigonometric polynomials of period 2L on [-L, L] are finite linear combinations of the mutually orthogonal plane waves [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [absolute value of [k.sub.m]] [less than or equal to] [OMEGA], it follows that the [f.sub.n] are converging in norm on [L.sup.2][-L, L] to a function orthogonal to the subspace of [OMEGA]-bandlimited trigonometric polynomials of period 2L on [-L, L]. That is, for any interval [-L, L] there exist [OMEGA]-bandlimited functions whose projections onto [L.sup.2][-L, L] are arbitrarily close to being orthogonal to the subspace [B.sub.L]([OMEGA]) [subset] [L.sup.2][-L, L] of [OMEGA]-bandlimited trigonometric polynomials of period 2L. Even more dramatically, for any interval [-L, L] there exist [OMEGA]-bandlimited functions which are in fact orthogonal to the subspace [B.sub.L]([OMEGA]) on [-L, L]. This will be demonstrated in Section 1.2.

This may appear to suggest that there are [OMEGA]-bandlimited functions which cannot be seen as a limit of [OMEGA]-bandlimited trigonometric polynomials. However, as shown in [7], superoscillatory behaviour comes at a price. Superoscillations are 'expensive in norm'. Roughly speaking, the amplitude of the superoscillations in a bandlimited signal is very small relative to the amplitude of the signal outside of the superoscillating interval, and an increase in the length of the superoscillating interval or the rapidity of the superoscillations corresponds to a large increase in the norm of the signal. As shown in [7], for a large class of superoscillatory [OMEGA]-bandlimited signals, if the norm of the bandlimited signal is fixed, the amplitude of superoscillations decreases polynomially with their wavelength, and if the wavelength of the superoscillations is also fixed, their amplitude decreases exponentially with the size of the superoscillating interval. This suggests that for a fixed bandlimited function with a fixed, finite norm, there is an upper bound on how much it can 'superoscillate'.

The results of this paper support this. Namely, for any fixed [OMEGA]-bandlimited function, regardless of how wild its local behaviour is in a given finite interval, if one views the function on a sequence of intervals of increasing length L, there exist [OMEGA]-bandlimited trigonometric polynomials on these intervals belonging to 2[??][OMEGA]L/[pi][??] + 1 dimensional subspaces of [L.sup.2][-L, L] which become arbitrarily good approximations to the original bandlimited function in the limit as L [right arrow] [infinity].

1.1 Trigonometric Polynomial Approximations of Bandlimited Functions

An [OMEGA]-bandlimited trigonometric polynomial p of period 2L is a linear combination of the plane waves [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [k.sub.n] := n[pi]/L for which [absolute value [k.sub.n]] [less than or equal to] [OMEGA]. If [absolute value [k.sub.n]] then this implies that [absolute value of n] [less than or equal to] [OMEGA]L/[pi]. Let N := [??][OMEGA]L/[pi][??]; then any such trigonometric polynomial p can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

The set of all [OMEGA]-bandlimited trigonometric polynomials of period 2L forms a 2[??][OMEGA]L/[pi][??] + 1 dimensional subspace [B.sub.L]([OMEGA]) of [L.sup.2][-L, L]. Any [OMEGA]-bandlimited trigonometric polynomial of period 2L is clearly holomorphic on the entire complex plane.

Let B([OMEGA]) denote the Hilbert space of functions bandlimited by [OMEGA]. The Fourier transform of any element of this Hilbert space is an clement of [L.sup.2] [-[OMEGA], [OMEGA]]. Given a bandlimited function f [member of] B([OMEGA]), consider its Fourier series on an interval [-L,L]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where [k.sub.n] := n[pi]/L and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now truncate the Fourier series of f on this interval to remove all complex exponential terms [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then N this truncated Fourier series can be written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where N = [??][OMEGA]L/[pi][??]. This resulting trigonometric polynomial,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where z [member of] C will be called the [OMEGA]-bandlimited trigonometric polynomial ([TP.sub.[OMEGA]]) version of f on the interval [-L, L]. It will be convenient to also consider the functions [[pi].sub.N] := [f.sub.N[chi]L] where [chi]L is the characteristic function of the vertical strip [absolute value of Re(z)] [less than or equal to] L (i.e.,) [chi]L is 1 on this strip and vanishes outside of it). The functions [[phi].sub.N] will be called the L-truncated [TP.sub.[OMEGA]] versions of f. These functions are analytic on the vertical strip [absolute value of Re(z)] < L.

These L-truncated [TP.sub.[OMEGA]] versions of a bandlimited function clearly belong to the Hilbert spaces of [OMEGA]-bandlimited trigonometric polynomials [B.sub.L]([OMEGA]) [subset] [L.sup.2][-L, L]. Let--[[DELTA].sub.L] denote the operator on [L.sup.2](R) which acts as the self-adjoint Laplacian (i.e., minus the second derivative operator) with periodic boundary conditions on [L.sup.2] [-L, L], and as the zero operator on the orthogonal complement of [L.sup.2][-L,L] in [L.sup.2](R). Further let [chi]L denote the projection of [L.sup.2](R) onto [L.sup.2][-L, L]. Then the L-truncated [TP.sub.[OMEGA]] version of a bandlimited function f [member of] B([OMEGA]) is simply the image of f under the projection operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2) This is clear since the eigenfunctions of -[[DELTA].sub.L] which have support only on [-L, L] are just the plane waves [ILLUSTRATION OMITTED] truncated to the interval [-L, L] with eigenvalues [k.sup.2.sub.n]. Further notice that [B.sub.L]([OMEGA]) is a natural generalization of the space of [OMEGA]-bandlimited functions to the spatially-limited, finite interval [-L, L] since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where--[DELTA] is the Laplacian or minus the second derivative operator on [L.sup.2](R).

Since a bandlimited function f [member of] B([OMEGA]) contains no frequencies larger in magnitude than [OMEGA], one may intuitively expect that a [TP.sub.[OMEGA]] version of f on an interval [-L, L] will become an increasingly better approximation to f as L (and therefore as N) approaches infinity. The results of the following sections will justify this intuition.

1.2 Proof that [B.sub.L]([OMEGA]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Before proceeding to state the main results of this paper, it will first be proven that [B.sub.L]([OMEGA]) is actually the image of B(A) under the spectral projection operator [ILLUSTRATION OMITTED] for any [LAMBDA] > 0. This will prove that any [OMEGA]-bandlimited trigonometric polynomial can be seen as the truncated [TP.sub.[OMEGA]] version of [OMEGA]-bandlimited function. It will further be shown that there exist functions f [member of] B([OMEGA]) for which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in [L.sup.2][-L,L]. Here [chi]L denotes the projection operator onto the subspace [L.sup.2][-L, L] of [L.sup.2](N). The method of proof used here is very similar to that used to construct superoscillating bandlimited functions in [7].

It needs to be shown that given any 2N + 1 complex values [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and any [LAMBDA] > 0 there exists an f [member of] B([LAMBDA]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

for all [absolute value of n] [less than or equal to] N where N := [??][OMEGA]L/[pi][??] and [k.sub.n] := n[pi]/L. This will show [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], proving that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. That [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is obvious.

[LAMBDA] [LAMBDA]--bandlimited function of minimum norm satisfying the 2[??][OMEGA]L/[pi][??] + 1 constraints (5) will now be constructed using variational methods. The Fourier transform of f [member of] [L.sup.2](R) is defined to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

The functional to be extremized can be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Setting the functional derivative of [PHI] to zero yields the Euler-Lagrange equation:

F(w) + [N.summation over (-N)] [[lambda].sup.*.sub.n] sin L(w - [k.sub.n])/L(w - [k.sub.n]) = 0. (8)

Using equation (5) this becomes

[a.sub.j] = -[N.summation over (n=-N)] [[lambda].sup.*.sub.N] [[integral].sup.[LAMBDA].sub.-[LAMBDA]] (sin L(w - [k.sub.j])/L(w - [k.sub.j])) (sin L(w - [k.sub.n])/L(w - [k.sub.n])) dw. (9)

Define

[S.sub.jn] =: [[integral].sup.[LAMBDA].sub.-[LAMBDA]] (sin L(w - [k.sub.j])/L(w - [k.sub.j])) (sin L(w - [k.sub.n])/L(w - [k.sub.n])) dw. (10)

This is a (2N + 1) x (2N + 1) matrix.

Given any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on [-L, L] and vanishes outside of [-L,L]. Now consider the norm of P[(w).sub.[chi][-[LAMBDA], [LAMBDA]] where P(w) is the Fourier transform of p(x) and [chi][-[LAMBDA], [LAMBDA] is the characteristic function of the interval [-[LAMBDA], [LAMBDA]],

P(w) = L/[pi] [L/[pi] [N.summation over (-N)] [p.sub.n] sin L(w - [k.sub.n])/L(w - [k.sub.n]) (11)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Note that since p(x) vanishes outside of [-L, L] that P(w) is holomorphic on the entire complex plane. Therefore, if p(x) [not equal to] 0, P(w) can only be zero on a discrete number of points in [-[OMEGA], [OMEGA]] which have no limit point so that [[integral].sup.[LAMBDA].sub.-[LAMBDA]] [[absolute value of P(w)].sup.2]dw must be positive. Thus, equation (12) shows that S has trivial kernel mid, is therefore, invertible.

Using the inverse of S in equation (9) now yields

[[lambda].sup.*.sub.k] = - [N.summation over (j=-N)] [S.sup.-1.sub.kj] [a.sub.j], (13)

Calculating the Lagrange multipliers [[lambda].sub.k] and substituting them into equation (8) then yields the Fourier transform of the desired [LAMBDA]-bandlimited function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

In summary given any 2N + 1 complex values {[a.sub.n]} one can explicitly construct a bandlimited function f [member of] B([LAMBDA]) such that its Fourier coefficients [f.sub.n] = [a.sub.n]

for all n [member of] {- N, ...,N}. In other words, given any p [member of] [B.sub.L]([OMEGA]) and [LAMBDA] > 0 there is an f in B([LAMBDA]) for which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Furthermore, the above result also shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any [LAMBDA] > [OMEGA]. Suppose [LAMBDA] is chosen such that M := [??][LAMBDA]L/[pi][??] > [??][OMEGA]L/[pi][??] = N. Then, given the sequence [{[a.sub.n]}.sup.M.sub.n=-M] where [a.sub.M] = 1, [a.sub.n] = 0 for n [not equal to] M, there exists an f [member of] B([OMEGA]) for which its Fourier coefficients on [-L, L] obey [f.sub.n] = [a.sub.n] for all n [member of] { - M, ..., M}. It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] . This shows that there exist [OMEGA]-bandlimited fimctions which are orthogonal to all [OMEGA]-bandlimited trigonometric polynomials on [- L, L] for any L > 0.

2 Statement of Main Results

Proposition 1. Any strictly [OMEGA]-bandlimited function f is the limit of any sequence of truncated [TP.sub.[OMEGA]. versions of itself, {[[phi].sub.N]}.sup.[infinity].sub.N = 1] where convergence is with respect to the norm of [L.sup.2](-[infinity], [infinity]).

A function is strictly bandlimited by [OMEGA] if it is bandlimited by A where [LAMBDA] is strictly less then [OMEGA]. Note that there is some freedom in the choice of the sequence {[[phi].sub.N]}. Since N = [??][LAMBDA]L/[pi][??] one is free to choose [[phi].sub.N] to be the L-truncated TP[OMEGA] version of f on any interval [- L, L] where L [member of] [[pi]/[OMEGA] N, [pi]/[OMEGA] (N + 1)). It is assumed that [OMEGA] is fixed so that as N [right arrow] [infinity], L [right arrow] [infinity]. In terms of the projection operators [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] discussed in the previous section, Proposition 1 can be rephrased in the following way. The operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges strongly to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the limit as L [right arrow] [infinity] for any [LAMBDA] < [OMEGA]. Since - [DELTA] has purely continuous spectrum, it is not difficult to show that this implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges strongly to zero as L [right arrow] [infinity].

This proposition will be used to establish the following stronger result.

Proposition 2. Given any strictly [OMEGA]-bandlimited f, any sequence of its truncated [TP.sub.[OMEGA] versions {[[phi].sub.N]}.sup.[infinity].sub.N = 1] converge to f in [L.sup.2] norm on any line parallel to the real axis in the complex plane. Furthermore, this [L.sup.2] convergence is uniform in any strip parallel to the real line in the complex plane.

By the [L.sup.2] convergence being uniform, it is meant that given any horizontal strip Im(z) [less than or equal to] B, and any [member of] > 0, there is an N' [member of] N such that for all N > N', [parallel] f - [[phi].sub.N][parallel]y < [member of] for all [absolute value of y] [less than or equal to] B. Here [parallel] f [parallel][sup.2.sub.y] = [[integral].sup.[infinity].sub.-[infinity][absolute value of f(x + iy).[sup.2]dx.

The following corollary is a straightforward application of Proposition 2.

Corollary 1. Given a strictly [OMEGA]-bandlimited function f, any sequence [{[[phi].sub.N]}.sup.[infinity].sub.N = 1] where N := [??][OMEGA]L/[pi][??] and [[phi].sub.N] is a truncated [TP.sub.[OMEGA] version of f on [- L, L] converges uniformly to f on any strip parallel to R in the complex plane.

Note that this corollary implies the weaker result that the sequence {fN}N[member of]N converges uniformly to the original strictly bandlimited f on any compact subset of C.

A set of points [LAMBDA] := {yn}n[member of]Z] is called uniformly discrete if [absolute value of yn - ym] [greater than or equal to] [member of] for some [member of] > 0 and all n, m [member of] Z. In this paper it is assumed that A C R.

Corollary 2. Suppose f is a strictly bandlimited function with bandlimit [OMEGA]. If [{[[phi].sub.N]}.sup.[infinity].sub.N = 1] where N := [??][OMEGA]l/[pi][??] is any sequence of truncated [TP.sub.[OMEGA] versions of f and [LAMBDA] := {yn}n[member of]Z is a uniformly discrete set of points, then the square summable sequence {[[phi].sub.N](yn)} converges to the sequence {f(yn)}n[member of]Z in [l.sup.2](Z).

3 Proof of Results

The following basic facts and inequalities for bandlimited functions will be useful in establishing the results.

The Fourier transform f [member of] B([OMEGA]) of F [member of] [L.sup.2][-[OMEGA], [OMEGA]] will be defined as

f(x) = [[integral].sup.[OMEGA].sub.-[OMEGA] F(w)[e.sup.iwx]dw x [member of] R. (15)

With this asymmetric definition the Fourier transform is not an isometry. If f [member of] B([OMEGA]) is the transform of F [member of] [L.sup.2][-[OMEGA], [OMEGA]], then [parallel]f[parallel] [square root of 2[pi][parallel]\f[parallel].

The space B([OMEGA]) is closed under differentiation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Here [f.sup.(n) denotes the nth derivative of f. Pointwise evaluation at any point in C is a bounded linear functional on B([OMEGA]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

Here y = Im(z). Thus,

[absolute value of [f.sup.(n)](z)] [less than or equal to] [[OMEGA].sup.n] [square root of [OMEGA]/[pi]][e.sup.[OMEGA][absolute value of y]][parallel]f[parallel]. (18)

Similarly, if [f.sub.N] is the [TP.sub.[OMEGA] version of f on [-L, L], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Using the fact that N = [??][OMEGA]L/[pi][??] it follows that 2N + 1 [less than or equal to] 2[OMEGA]L/[pi] + 1. Therefore, for all N,

[[absolute value of [f.sup.(j).sub.N](z)].sup.2] [less than or equal to] [e.sup.2[OMEGA] [absolute value of y]][[OMEGA].sup.2j] ([OMEGA]/[pi] + 1/2L) 2L [N.summation over (n=1N)] [[absolute value of fn].sup.2]. (20)

Using the fact that 2L [[SIGMA].sup.N.sub.n]= N [[absolute value of fn].sup.2] [less than or equal to] 2L [[SIGMA].sup.[infinity].sub.n=-[infinity] [[absolute value of fn].sup.2] = [[integral].sup.L.sub.-L][[absolute value of f(x)].sup.2]dx [less than or equal to] [parallel]f[parallel][sup.2] for all N [member of] N, it follows that

[absolute value of [f.sup.(j).sub.N](z)] [less than or equal to] C[[OMEGA].sup.j][e.sup.[OMEGA][absolute value of y]] [parallel]f[parallel] (21)

where [c.sup.2] := ([OMEGA]/[pi] + 1/2L).

3.1 Proof of Proposition 1

The following basic facts about uniform convergence and interchanging limits will be needed.

Theorem 1. (Weierstrass M-test) Let [{fn}.sup.[infinity].sub.n = 1] be a sequence of functions defined on a set E. Suppose that [absolute value of fn(x)] [less than or equal to] [M.sub.n] for all x [member of] E and all n [member of] N. Then the sequence of partial sums [[SIGMA].sup.N.sub.n=1] fn(x) converges uniformly on E if [[SIGMA].sup.[infinity].sub.n=1] [M.sub.n] converges.

Theorem 2. (theorem 7.11 pg 149 [14]) Suppose that fn [right arrow] f uniformly on E. If t is a limit point ore and [lim.sub.x[right arrow]t] fn(x) = [A.sub.n] for every n [member of] N then [{[A.sub.n]}.sup.[infinity].sub.n=1] converges and [lim.sub.x[right arrow]t] f(l) = [lim.sub.n[right arrow][infinity] [A.sub.n], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using these two theorems it is straightforward to establish the following fact that will be used in the proof of Proposition 1.

Theorem 3. Suppose [{fnm}.sup.[infinity].sub.n,m=1] is a doubly infinite sequence of functions defined for all x [member of] E and [absolute value of fnm(x)] [less than or equal to] [M.sub.nm] for all x [member of] E where [[SIGMA].sup.[infinity].sub.n,m=1] [M.sub.nm] < [infinity. Then if t is a limit point of E,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It will be convenient to first prove a few lemmas before establishing Proposition 1.

Lemma 1. Let N := [??][OMEGA]L/[pi][??]. If [absolute value of n] > N then the nth Fourier coefficient in the Fourier series of a function bandlimited by [OMEGAA] on the interval [-L, L] is given by the formula

FN := [(-1).sup.(n + 1)]/2L [[infinity].summation over (j=1)] [f.sup.(j-1)] (L) - [f.sup.(j-1)] (- L)/[[(ik.sub.n]).sup.j].

In other words, the Fourier coefficient of any plane wave in the Fourier series of f [member of] B([OMEGA]) on [-L, L] with frequency > [OMEGA] is given by the above formula.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Integrating by parts yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

Repeatedly integrating by parts gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Define [S.sub.j] := [(-1).sup.n+1] [[summation].sup.j.sub.m=1] [f.sup.(m -1) (L) - [f.sup.(m - 1)] (- L)/[([ik.sub.n]).sup.m]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

Assume without loss of generality that [parallel] f [parallel] = 1. Using the bound (18) on the jth derivative of f at any point on the real line, equation (25) becomes

[absolute value of 2L[f.sub.n] - [S.sub.j]] [less than or equal to] [square root of [OMEGA]/[pi]] [([OMEGA]/[absolute value of [k.sub.n]]).sup.j][[integral].sup.L.sub.-L] dx = [square root of [OMEGA]/[pi]] 2L [([OMEGA]/[absolute value of [k.sub.n]]).sup.j]. (26)

However, since [absolute value of [k.sub.n] > [OMEGA] for all [absolute value of n] > N it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

establishing the claim.

Lemma 2. The functions [g.sub.n](x) = [x.sup.n][(-1).sup.(n+1)][[??].sub.n](x) are monotonically decreasing functions of x for x > O, for every n [member of] N. Here [[??].sub.n](x) = [d.sup.nnnn+1]/[dx.sup.n+1] ln [GAMMA] (x) is the nth polygamma function

The proof of this lemma is straightforward if one uses the fact [1] that

[[??].sub.n](x) = [(-1).sup.n+1] [[integral].sup.[infinity].sub.0] [t.sup.n][e.sup.-xt]/1 - [e.sup.-t] dt. (28)

Now Proposition 1 will be proven.

Proof. (of Proposition 1) Suppose f is bandlimited by [LAMBDA] < [OMEGA], and assume without loss of generality that [parallel]f[parallel] = 1. Let [phi]N be the truncated [TP.sub.[OMEGA] version of f on an interval [-L, L], N = [??][OMEGA]L[[pi][??].

Then

[parallel]f - [phi]N[parallel][sup.2] = [[integral].sub.[absolute value of x]>L [[absolute value of f(x)].sup.2]dx + 2L [summation over ([absolute value of n]>N]] [absolute value of [f.sub.n]].sup.2]. (29)

The coefficient [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the n-th coefficient in the Fourier series of f on [-L, L]. As N [right arrow] [infinity], L [right arrow] [infinity] and the integral in (29) vanishes in this limit as f is square integrable. Thus, to show that [phi]N converges to f as N [right arrow] [infinity] one needs to show only that [lim.sub.[L [right arrow][infinity] 2L [{summation].sub.[absolute value of n] > N] [[absolute value of fn].sup.2] = 0. It will be proven that [lim.sub.L [right arrow][infinity] 2L [{summation].sub.n > N] [[absolute value of fn].sup.2] = 0. Showing that the limit of the other half of the sum vanishes uses the exact same logic.

Let S(L) := 2L [[SIGMA].sup.[infinity].sub.n=N + 1] [[absolute value of fn].sup.2]. Using the formula from Lemma 1 this becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

It will be useful to interchange the orders of summation in S. To show that this is valid, it must be shown that S converges absolutely, i.e., it must be shown that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

for any fixed value of L > 0. Now T is hounded by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Using the fact that [absolute value of [f.sup.(j)] {x}] [less than or equal to] [square root of [OMEGA]/[pi]] [[OMEGA].sup.j] [parallel]f[parallel] = [square root of [OMEGA]/[pi]] [[OMEGA].sup.j] as shown in equation (18), it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

Again since [k.sub.n] > [OMEGA] for all [absolute value of n] > N the above sums in j and r are convergent geometric series, and are easily evaluated to give the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

Now N := [??][OMEGA]L/[pi][??] so that N + 1 = [OMEGA]L/[pi] + [delta] where 1 [greater than or equal to] [delta] > 0. Letting n = s + N + 1 = (s + [delta]) + [OMEGA]L/[pi] and using that [k.sub.n] = n[pi]/L the sum in (34) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)

This shows that S(L) converges absolutely for any fixed L so that the orders of summation can be interchanged. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

The sum farthest to the right can be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

The nth polygamma function can be expressed as [[psi].sub.n](z) = [(-1).sup.n+1]n! [[summation].sup.[infinity].sub.k=0] 1/[(z + k).sup.n + 1] [1]. Using the polygamma functions, [??] can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

Substituting this expression into that for [S.sub.jr] (L) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

To show that the limit as L [right arrow] [infinity] of the double sum S(L) = [[summation].sup.[infinity].sub.j,r=1] [S.sub.jr](L) vanishes it will be shown that the conditions of Theorem 3 are satisfied so that this limit can be interchanged with the summations. That is, it will be shown that [S.sub.jr](L) [less than or equal to] [M.sub.jr] for all L [member of] [[pi]/[OMEGA], [infinity]) where [[summation].sup.[infinity].sub.j,r=1] [M.sub.jr] < [infinity].

By Lemma 2 the expression [g.sub.j+r-1]([[OMEGA]L/[pi] := [([OMEGA]L/[pi]).sup.j+r-1] [(-1).sup.j+r] [[psi].sub.j+r-1] ([OMEGA]L/[pi]) is a monotonically decreasing function of [OMEGA]L/[pi] for all [OMEGA]L/[pi] > 0. Hence, for fixed [OMEGA] and all L > 0 this expression is a monotonically decreasing function of L. In particular, [g.sub.j+r-1]([OMEGA]L/[pi]) [less than or equal to] [g.sub.j+r-1](1) = [(-1).sup.j+r][[psi].sub.j+r-1](1) for all L in the interval [[pi]/[OMEGA], [infinity]). Using this fact and that [absolute value of [f.sup.(j)(x)]] [less than or equal to] [square root of [LAMBDA]/[pi]][[LAMBDA].sup.j] since f is bandlimited by [LAMBDA] < [OMEGA], the summand [S.sub.jr](L) is bounded by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

for all L [member of] [[pi]/[OMEGA], [infinity]. Finally, using the identity [[psi].sub.n](1) = [(- 1).sup.n+1]n![zeta](n+1) [1] for all n [member of] N and the fact that the Riemann zeta function [zeta](n) = [[summation].sup.[infinity].sub.k=1] 1/[k.sub.n] is clearly monotonically decreasing for all n > 1, we have that [[psi].sub.n](1) [less than or equal to] [(- 1).sup.n+1]n![zeta](2)= [(-1).sup.n+1]n![[[pi].sup.2]/6 for all n [greater than or equal to] 1. Therefore,

[(-1).sup.j+r][[psi].sub.j+r-1](1)/(j + r - 1)! [less than or equal to] [[pi].sup.2]/6 (41)

so that

[S.sub.jr](L) [less than or equal to] 1/3 [([LAMBDA]/[OMEGA]).sup.j+r-1] =: [M.sub.jr] (42)

for all L [member of] [[pi]/[OMEGA], [infinity]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (43)

This shows that the conditions of Theorem 3 are satisfied so that the limit as L [right arrow] [infinity] can be interchanged with the double sum:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (44)

This limit is zero since each [f.sup.(j)] is bandlimited so that [lim.sub.L [right arrow][infinity]] [absolute value of [f.sup.(j)]([+ or -] L)] = 0. Since S [greater than or equal to] 0, this shows that [lim.sub.L[right arrow][infinity]] S = 0.

The above proposition can be applied to any bandlimited function by noting that, if f is bandlimited by [OMEGA], it is strictly bandlimited by any [GAMMA] = [OMEGA] + [member of] where [member of] > 0 is arbitrary.

3.2 Proof of Proposition 2

The following theorems will be needed in the proof of this proposition.

Theorem 4. (theorem 12 pg. 83 [17]) If f is an entire function of exponential type and if f(x) [arrow right] 0 as [absolute value of x] [right arrow] [infinity], then f(x + iy) [right arrow] 0 as [absolute value of x] [right arrow] [infinity] uniformly in every horizontal strip.

Theorem 5. (Plancherel - Polya) If g is an entire function of exponential type [OMEGA] and if for some p > 0,

[[integral].sup.[infinity].sub.-[infinity]] [[absolute value of g(x))].sup.p]dx < [infinity]

then

[[integral].sup.[infinity].sub.-[infinity]] [[absolute value of g(x + iy)].sup.p]dx [less than or equal to] [e.sup.p[OMEGA][absolute value of y]] [[integral].sup.[infinity].sub.-[infinity]] [[absolute value of g(x)].sup.p]dx.

Define gb(z) = f(z + ib) where b is a constant real number and f is strictly bandlimited by [OMEGA]. Then gb is an entire function since f is. Let gb,N(z) be the [OMEGA] bandlimited TP version of gb on [-L,L], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

where N := [??][OMEGA]L/[pi][??] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] . Finally let [gamma]b,n(z) : = XL(z)gb,N(z).

Lemma 3. Given gb as described above, any sequence of the truncated [TP.sub.[OMEGA]] functions [{[gamma]b,N}.sup.[infinity].sub.N=1] converges to gb with respect to the [L.sup.2] norm on R. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (46)

Furthermore, this [L.sub.2] convergence is uniform on any strip parallel to the real line, i.e., [for all][absolute value of b] [less than or equal to] B where B > 0 is fixed, in the same sense as described following the statement of Proposition 2.

Since gb is simply a vertical translation of the strictly bandlimited function f, an equality similar to (18) holds for gb:

[absolute value of [g.sup.(j).sub.b](z)] = [[absolute value of [f.sup.(j)](z + iB)].sup.2] [less than or equal to] [parallel]f[parallel][e.sup.[OMEGA][absolute value of B + y] [square root of [OMEGA]/[pi]][[OMEGA].sup.j]. (47)

Since the proof of Lemma 3 is very similar to that of Proposition 1, its proof will be merely sketched here.

Using the bound (47), it is straightforward to verify that the following formula directly analogous to the one proven in Lemma 1 holds for all [absolute value of n] > [??][OMEGA]L/[pi][??]:

gb,n = [(-1).sup.n+1]/2L [[infinity].summation over (j=1) 1/([[ik.sub.n]).sup.j] ([f.sup.(j-1)] (L + ib) - [f.sup.(j- 1)] (-L + ib)). (48)

Now the difference in norm between gb and [gamma]b,N is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

By Theorem 5, on any fixed horizontal strip [absolute value of Im(z)] [less than or equal to] B in the complex plane, the integrals [[integral].sub.[absolute of x]>L] [[absolute of gb(x)].sup.2]dx converge to zero uniformly as L [arrow right] [infinity]. Repeating the same steps as in Proposition 1 then leads to the following equation, similar to equation (45):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (50)

By Theorem 4 it follows that this quantity also converges uniformly to zero for all [absolute value of b] [less than or equal to] B for any fixed B.

Proposition 2 will now be proven. For convenience, let [[parallel]f[parallel].sup.2.sub.y] := [[integral].sup.[infinity].sub.-[infinity]] [[absolute value of f(x + iy)].sup.2]dx.

Proof. Let B > 0 be arbitrary. To show that the sequence of truncated [TP.sub.[OMEGA]] functions {[[pi].sub.N]} converge in norm to f on any line x + ib parallel to the real axis, uniformly for all [absolute value of b] [less than or equal to] B, it suffices to show that [f.sup.[infinity].sub.- [infinity] [absolute value of [pi].sub.N](x + ib)-[gamma]b,N(x)][sup.2]dx [right arrow] 0 as N [right arrow] [infinity] uniformly for [absolute value of b] [less than or equal to] B where [gamma]b,N is the truncated [OMEGA]-bandlimited TP version of gb(z) = f(z + ib). This follows since by the previous lemma,

[[integral].sup.[infinity].sub.-[infinity] [[absolute value of f(x + ib) - [gamma]b,N(x)].sup.2]dx [right arrow] 0 (51)

uniformly for [absolute value of b] [less than or equal to] B, as N [right arrow] [infinity]. Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (53)

Recall that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (54)

Now consider a counterclockwise oriented rectangular contour S in the complex plane with vertices (-L, 0), (L, 0), (L, b), and (-L, b). The function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is entire, and so by Cauchy's theorem [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (55)

Substituting this into equation (54) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (56)

Substituting this inequality (56) into equation (53) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

This same upper bound holds for all [absolute value of b] < B. Since f is bandlimited, Theorem 4 implies that this goes to zero uniformly for all [absolute value of b] [less than or equal to] B in the limit as L (or equivalently as N) goes to infinity.

Using this result, it is straightforward to establish uniform pointwise convergence on any horizontal strip in the complex plane.

3.3 Proof of Corollaries

Proof. (of Corollary 1) Assume the contrary. That is, assume that there is a horizontal strip S := {z [member of]C | [absolute value of Im(z)] [less than or equal to] B} Oil which the sequence {[[phi].sub.N]} of truncated [OMEGA]-bandlimited TP versions of a strictly bandlimited f [member of] B([OMEGA]) do not converge Uniformly to f. Then there exists a number [epsilon] > 0 such that for every N [member of] N there is an N' > N and a point [z.sub.N'] = [x.sub.N'] + [iy.sub.N'] [member of] S for which [absolute value of f([z.sub.N']) - [[phi].sub.N']([z.sub.N'])] > 2[epsilon]. Using the bounds (21) and (18) it follows that

[absolute value of f'(z) - [[phi]'.sub.N'](z)] [less than or equal to][absolute value of f'(z)] + [absolute value of [[phi]'.sub.N'](z)] [less than or equal to] K [parallel]f[parallel] : = M (58)

for all z [member of] S, where M < [infinity]. Now choose [??] := [??][OMEGA][??]/[pi][??] so large that for [absolute value of x] > [??], [absolute value of f(x + iy)] < [epsilon] for all [absolute value of y] [less than or equal to] B. This can be done by Theorem 4. This shows that for all N > [??], if [absolute value of x] > [L.sub.N] and x + iy [member of] S then [absolute value of f(x + iy) - [[phi].sub.N](x + iy)] = [absolute value of f(x + iy)] < [epsilon]. Therefore, the points [z.sub.N], lie in the rectangles [absolute value of x] [less than or equal to] [L.sub.N'], [absolute value of y] [less than or equal to] B for all N' > [??].

Now the difference function [g.sub.N'] = f - [[phi].sub.N'] is analytic for all z = x + iy such that [absolute value of x] < L' and continuous for [absolute value of x] [less than or equal to] L'. Consider a point w = x + [iy.sub.N'] [member of] S for which [absolute value of w - [z.sub.N']] < [epsilon]/M.

Now consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (59)

where the bound (58) was used.

Therefore,

[epsilon] [absolute value of [g.sub.N'](w) - [g.sub.N']([z.sub.N'])] [greater than or equal to][absolute value of [g.sub.N']([z.sub.N']] - [absolute value of [g.sub.N'](w)] > 2[epsilon] [absolute value of [g.sub.N'](w)] (60)

so that [absolute of gN'(w)] > [epsilon] for all w = x + [iy.sub.N'] such that [absolute value of x - [x.sub.N']] < [epsilon]/M. It then follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (61)

In conclusion, for every N [member of] N there is an N' > N for which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [greater than or equal to] [[epsilon].sup.2]/2M] > 0 where [absolute value of [y.sub.N']] [less than or equal t o] B for all N'. This is a contradiction since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] goes to zero uniformly for all y [less than or equal to] B in the limit as N [right arrow][infinity] by Proposition 2.

Proposition 2 can again be applied to prove Corollary 2. The proof of this corollary is very similar to that of Theorem 17 in [17].

Proof. (of Corollary 2) Let f be strictly bandlimited by [OMEGA] and let [{f N}.sup.[infinity].sub.N=1] be a sequence of [OMEGA] bandlimited TP versions of f. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a uniformly discrete set of real points. Then [LAMBDA] has no limit points, and there exists an [epsilon] > 0 such that [absolute value of [y.sub.n] - [y.sub.m]] [greater than or equal to] [epsilon] for all n, m [member of] Z.

Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (62)

Recall that [[phi].sub.N] = [chi]L[florin]N. Since the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is square summable, the first stun in equation (62) will vanish in the limit as N (or equivalently L) approaches infinity. Therefore, to prove the corollary it needs to be shown that the second sum also vanishes in this limit.

Let [g.sub.N] := [f.sub.g] - f. Since [f.sub.N] - f is entire, [[absolute value of [g.sub.N]].sup.p] is subharmonic for any p [member of] N. Given any [z.sub.0] in C it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (63)

For a description of subharmonic functions and their properties, see for example [13]. Multiplying both sides by r, integrating from 0 to [delta], and then switching from polar to cartesian co-ordinates gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (64)

Here x = Re(z) and y = Im(z).

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (65)

Letting [delta] = [epsilon]/2, the integrals in the above sum (65) become pairwise disjoint so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (66)

By Proposition (2) the first double integral in equation (66) converges to 0 in the limit as N approaches infinity. The second double integral can be bounded in the following way. First observe that the triangle inequality implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (67)

In the last line the fact that [f.sub.N] is the 2L periodic version of [[phi].sub.N] was used. Using Theorem (4) and Proposition (2), it is not difficult to see that this vanishes in the linfit as N (and L) approach infinity, uniformly for [absolute value of y] [less than or equal to] [delta]. It follows that the second double integral in the last line of equation (66) vanishes in this same limit, proving the claim.

Corollary 2 and Proposition 1 immediately imply the following sufficiency and necessity conditions for a uniformly discrete set of real points [LAMBDA] to be a set of sampling for B([OMEGA]).

Corollary 3. Let f, [LAMBDA], and the sequence {[[phi].sub.N]} be as in the previous corollary. Suppose that there is an N' [member of] N such that for all N > N',

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (68)

where 0 < b [less than or equal to] B < [infinity] are independent of the choice of strictly bandlimited [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a set of sampling for B([OMEGA]).

Conversely suppose there exists a non-zero strictly bandlimited f [member of] B([OMEGA]) such that for any C > 0 there is an N' such that if N > N' then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (69)

Then [LAMBDA] is not a set of sampling for B([OMEGA]).

The sum in equation (68) contains only a finite number of non-zero terms since for each fixed N = [??][OMEGA]L/[pi][??} the functions [[phi].sub.N] vanish outside of [-L, L] and there is a smallest non-zero distance between any two elements of [LAMBDA].

Proof. By Proposition 1 the middle term of equation (68) converges to the norm squared of f as N [right arrow] [infinity], while the left and right hand sides converge to the square sum of the samples {f([[lambda].sub.n])} times B and b respectively in the same limit by Corollary 2. This proves that

B [summation over (n [member of] Z) [[absolute value of f([y.sub.n])].sup.2] [greater than or equal to] [[parallel]f[parallel].sup.2] [greater than or equal to] b [summation over (n [member of] Z) [[absolute value of f([y.sub.n])].sup.2] (70)

for all f [member of] B([OMEGA]) that are strictly bandlimited. Since strictly [OMEGA]-bandlimited functions are dense in B([OMEGA]) it is not difficult to show that the inequality (70) holds for all [florin] [member of] B([OMEGA]). The converse statement is similarly straightforward to establish.

4 Discussion

Corollary (3) of the previous section shows that uniformly discrete sets of points which are sets of sampling for the subspaces [B.sub.L]([OMEGA]) of [OMEGA]-bandlimited trigonometric polynomials for all L will also be sets of sampling for B([OMEGA]) provided certain conditions are satisfied. This relationship between the reconstruction and interpolation properties of [OMEGA]-bandlimited trigonometric polynomials and those of [OMEGA]-bandlimited functions will be discussed in more detail here.

4.1 Reconstruction and Interpolation

The study of the reconstruction and interpolation properties of [OMEGA]-bandlimited functions has been an active area. of research for many years. A set of points [LAMBDA} = [{[y.sub.n]}.sub.n[member of]Z] which has no limit points is called a set of sampling for the Hilbert space of [OMEGA]-bandlimited functions B([OMEGA]) if the norm squared of every bandlimited function is bounded above by the square sum of its samples taken on the points of [LAMBDA]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (71)

The lower inequality always holds if [LAMBDA] has no limit points (see ,e.g., [17] Theorem 17, pg. 97). Any bandlimited function can be stably reconstructed from its values taken on a set of sampling. The reconstruction is stable in the sense that a bounded error in the measurement of the samples of a bandlimited function (bounded with respect to the norm of [l.sup.2](Z)) can yield at most a bounded error in the reconstructed function (bounded with respect to the [L.sup.2] norm). A set of points [LAMBDA] with no limit points is called a set of interpolation if given any square summable sequence {[a.sub.n]} [member of] [l.sup.2](Z) there is a bandlimited function which takes the values of that sequence on the points of [lambda]. Finally the set [lambda] is called a set of uniqueness if no two different bandlimited functions take the same sample values on the points of [LAMBDA]. Although any bandlinfited function can be reconstructed everywhere from its values taken on a set of uniqueness, this reconstruction is generally not stable. (The linear operator which maps the square summable sample values that a bandlimited function takes on a set of uniqueness which is not a set of sampling onto the bandlimited function will be unbounded.)

Determining what properties a discrete set of real or complex values [LAMBDA} := [{[y.sub.n]}.sub.n[member of]Z] must possess in order to be a set of sampling, uniqueness or interpolation is in general very difficult, e.g., [10, 4]. On the other hand finding sets of sampling and interpolation for [OMEGA]-bandlimited trigonometric polynomials is easy. Interpolation and reconstruction of [OMEGA]-bandlimited trigonometric polynomials is straightforward and intuitive since they form a finite dimensional Hilbert space. Any element p of [B.sub.L]([OMEGA]) can be written as p(x) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [k.sub.n] := n[pi]/L and the sequence of complex numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is arbitrary. Since each element of [B.sub.L]([OMEGA]) has 2N + 1 'degrees of freedom', the 2N + 1 Fourier coefficient [{[p.sub.n]}.sup.N.sub.n=-N], one may intuitively expect that any p [member of] [B.sub.L]([OMEGA]) should be perfectly reconstructible from its values taken on any 2N + 1 values in the interval [-L, L). Indeed, it is straightforward to show that any element of [B.sub.L]([OMEGA]) has at most 2N zeros in the vertical strip [absolute value of Re(z)] [less than or equal to] L. A simple conformal map of the vertical strip [absolute value of Re(z)] [less than or equal to] L onto C which maps the line [-L, L] to the unit circle, and all horizontal lines to circles shows that every element of [B.sub.L]([OMEGA]) has no more zeros then a polynomial of degree 2N. It follows that the linear operator which maps the 2N + 1 Fourier coefficients [{[p.sub.n]}.sub.N.sub.n = -N] of p [member of] [B.sub.L]([OMEGA]) onto the 2N + 1 sample values [{p([y.sub.n])}.sup.N.sub.n=-N], where [{[y.sub.n])}.sup.N.sub.n=-N] are any 2N + 1 points in [-L, L), is invertible. Therefore, given any 2N + 1 points {[y.sub.n]} in I-L, L), any element of [B.sub.L]([OMEGA]) is perfectly reconstructible from the values it takes on those points, and given any 2N + 1 complex values {[a.sub.n]}, there is an element of [B.sub.L]([OMEGA]) that takes those values on the points [y.sub.n], p([y.sub.n]) = [a.sub.n]. In summary, any 2N + 1 points in the interval [-L, L) is both a set of sampling and a set of interpolation for [B.sub.L]([OMEGA]). Note that, since [B.sub.L]([OMEGA]) is finite dimensional, any set of uniqueness is also a set of sampling.

4.2 Condition Number of the Reconstruction Matrix

Corollary 3 provides some information on the relationship between sets of sampling for B([OMEGA]) and sets of sampling for [B.sub.L]([OMEGA]). For example, let [LAMBDA] := {[y.sub.n]} be a uniformly discrete set of real values such that for all N > N' [member of] N there are at least 2N + 1 points of [LAMBDA] in the interval [-[L.sub.N], [L.sub.N]) where [phi]N/[OMEGA] [less than or equal to] [L.sub.N] < [pi]/[OMEGA](N + 1). Then for each N > N' the set of points [LAMBDA][intersection] [-[L.sub.N], [L.sub.N]) is a set of sampling for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now consider the matrix M which maps the 2N + 1 sample values of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] onto the 2N + 1 Fourier coefficients, [{[[phi].sub.n].sup.N.sub.n=- N] of [phi]. Here

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (72)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (73)

It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Note that the Fourier coefficients are scaled differently here than in previous sections. Now suppose that [LAMBDA] satisfies the assumptions of the first part of Corollary 3 so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (74)

where b, B are independent of N > N'. This shows that the condition numbers (the ratio of the largest to the smallest eigenvalue) of the matrices M, which map the sample values [{[phi](y.sub.n])}.sup.N.sub.-N] onto the Fourier coefficients [{[[phi].sub.n]}.sup.N.sub.N] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], are bounded above by B/b < [infinity]. Thus, the first part of Corollary 3 can be restated in the following way. If [LAMBDA] is a uniformly discrete set of real points, and there exists N' [member of] N such that for all N > N', the condition numbers of the matrices which map the sample values of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] onto the Fourier coefficients of [phi] (as defined in equation (73)) are bounded above by some C < [infinity] then [LAMBDA] is a set of sampling for B([OMEGA]). In other words, if the condition numbers of these finite dimensional matrices which reconstruct elements of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from their samples taken on points of [LAMBDA] do not diverge in the limit as N [right arrow] [infinity], then [LAMBDA] is a set of sampling for B([OMEGA]). This is exactly what one would expect.

4.3 Outlook

The results of this paper that any [OMEGA]-bandlimited function can be seen as the limit of [OMEGA]-bandlimited trigonometric polynomials could provide a different perspective on the problem of determining whether or not a given uniformly discrete set of points is a set of sampling, uniqueness or interpolation for [OMEGA]-bandlimited functions. It will be an interesting avenue of future research to see whether facts about sets of sampling, uniqueness and interpolation for B([OMEGA]) which are more concrete then Corollary (3) can be derived from properties of [B.sub.L]([OMEGA]) and the theorems of this paper.

The proof of Proposition 1 showed that the operators ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converge strongly to zero as L [right arrow] [infinity]. Here -[[DELTA].sub.L] was the Laplacian with periodic boundary conditions on [-L, L] and the zero operator outside of this interval, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] was the projector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] was the projection of [L.sup.2](R) onto [L.sup.2][-L, L] and -[DELTA] was the Laplacian (minus the second derivative operator) on [L.sup.2](R). One would expect that the choice of boundary conditions on [-L, L] should not matter provided the resulting Laplacian is self-adjoint, but this remains to be checked. the author is currently generalizing the results of this paper to curved manifolds, where the analogue of [OMEGA]-bandlimited functions on curved manifolds is given in [12, 9]. This will be the subject of a future paper.

ACKNOWLEDGEMENT

The author would like to thank Dr. Achim Kempf for the support and discussions that stimulated this research. This work was partially funded by the Postgraduate Scholarship Program of the National Science and Engineering Research Council of Canada.

References

[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Dept. of Commerce, Washington D.C., 1972.

[2] Y. Aharonov, B. Reznik and A. Stern, Quantuln limitations on superluminal propagation, Phys. Rev. Lett., 81, 2190-2193, 1998.

[3] M.V. Berry, Evanescent and real waves in quantum billiards and Oaussian beams, J. Phys. A, 27, Lagl-L398, 1994.

[4] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Am. Math. Soc. 72:2, 341-366, 1952.

[5] K. Grochenig, Irregular sampling, Toeplitz matrices and the approximation of entire functions of exponential type, Math. Comp., 68:226, 749-765, 1999.

[6] M. S. Calder and A. Kempf, Analysis of superoscillatory wavefunctions, J. Math. Phys., 46, 01201, 2005.

[7] P.J.S.G. Ferreira and A. Kempf, Superoscillations: faster than the Nyquist rate, IEEE Trans. Signal Processing 54, 2006. In press.

[8] A. Kempf, Black holes, bandwidths and beethoven, J. Math. Phys., 46, 2360-2374, 2000.

[9] A. Kempf, A covariant information-density cutoff in curved spacetime, Phys. Rev. Lett., 92:22, 221301, 2003.

[10] H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Mathematica, 117, 37-52, 1967.

[11] L. Levi, Fitting a bandlimited signal to given points, IEEE Trans. Inform. Theory, 11,372-376, 1965.

[12] I. Pesenson, A sampling theorem on homogenous manifolds, Trans. Am. Math. Soc., 352:9, 4257-4269, 2000.

[13] W. Rudin Real and Complex Analysis, McGraw-Hill Inc., New York, 1966.

[14] W. Rudin Principles of Mathematical Analysis, McGraw-Hill Inc., New York, 1976.

[15] H.J. Schlebusch and W. Splettstosser, On a conjecture of J.L.C. Sanz and T.S. Huang, IEEE Trans. Acoust., Speech and Signal Processing 33:6, 1628-1630, 1985.

[16] D. Slepian, and H.O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty I, Bell Sys. Tech. J. 40:1, 43-63, 1960.

[17] R.M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press Inc., New York, 1980.

R. Martin

Applied Mathematics, University of Waterloo, 200 University Ave. W. Waterloo, ON, Canada

rtwmartin@math.uwaterloo.ca

(1) Note that this fact follows immediately from the results of Beurling which appear in [10] since any finite set of points is a set of interpolation for [OMEGA]-bandlimited functions.

(2) Here if A is a self-adjoint operator and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the characteristic function of [0, [[OMEGA].sup.2]], the spectral projection [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A) is defined by the functional calculus.
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Publication:Sampling Theory in Signal and Image Processing
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Date:Sep 1, 2007
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