# Global and local approximation behavior of reconstruction processes for Paley-Wiener functions.

Abstract

For signals in the Paley-Wiener space [PW.sup.1.sub.[pi]] a reconstruction in the form of a sampling series that is uniformly convergent on compact subsets of the real axis and uniformly bounded on the whole real axis is not possible in general if the signals are sampled equidistantly at Nyquist rate. We prove that, even if the signal is non-uniformly sampled with an average sampling rate equal to the Nyquist rate, a uniformly convergent reconstruction is not possible. Additionally, we provide a detailed convergence analysis and give a sufficient condition for the uniform convergence of the Shannon sampling series without oversampling. However, if oversampling is applied, a uniformly convergent reconstruction is always possible and as far as convergence is concerned no elaborate kernel design is necessary. Moreover, we show that a projection of the reconstruction process onto the range of signal frequencies is not possible without losing the good convergence behavior.

Key words and phrases : Sampling Theorem, Oversampling, Paley-Wiener Space, Signal Reconstruction, Uniform Convergence, Complete Interpolating Sequence, Riesz Basis

2000 AMS Mathematics Subject Classification 94A20, 94A05

1 Introduction

A reconstruction of signals from its samples by means of a sampling series that is uniformly convergent on compact subsets and uniformly bounded on the whole real axis is important not only from the theoretical point of view but in particular for practical applications. For the Paley-Wiener space [PW.sup.2.sub.[pi]] there is one certain sampling series that has become an integral part of modern communication and information theory. This sampling series is called Shannon sampling series, nowadays, although it has also been found by others, either earlier or independently at the same time. Whittaker introduced it in the mathematical literature in 1915 [27]. Several years later the sampling theorem appeared in the engineering literature: Kotel'nikov published it in 1933 [18], Raabe in 1939 [24] and Shannon in 1949 [26]. Since the work by Kotel'nikov was published in Russian and Shannon's paper was already written in 1940 [10], it is reasonable to assume that all three publications were created independently. See [10] and [20] for further historical details and [21, Chapter 2] for an overview of the topic. Recent developments in sampling theory can be found in [15, 14]. Chapter 3 of [21] deals with non-uniform sampling.

However, for practical applications the analysis of the space [PW.sup.2.sub.[pi]] is often not sufficient. Therefore, extensions in several directions have been considered, e.g., sampling theorems for more general function spaces [29, 11, 23, 17], sampling series with more general kernels (bandlimited and non-bandlimited) [14, Chapter 6] [7, 8, 12], multiband sampling and multidimensional sampling [14].

In this paper we do the analysis for the space [PW.sup.1.sub.[pi]], which is the largest space in the scale of Paley-Wiener spaces, because this space has some remarkable properties, e.g., the existence of the bandlimited interpolation. Therefore, this paper can be considered to be a continuation of the classical work by Brown and Butzer.

For signals f [member of] [PW.sup.1.sub.[pi]] there exists no uniformly convergent reconstruction process that uses only samples taken equidistantly at Nyquist rate, although this set of samples constitutes a set of uniqueness. It has been shown that this result is valid not only for the Shannon sampling series, but also for a fairly general class of reconstruction processes [2]. In this paper we consider non-equidistant sampling, with the sampling points [{[t.sub.k]}.sub.k[member of]Z] being real and satisfying [absolute value of [t.sub.k]-k] [less than or equal to] [delta] < 1/3, [member of] Z. We show that for a whole class of sampling series there are signals f [member] [PW.sup.1.sub.[pi]] such that neither global uniform convergence nor global boundedness is given. The question is: how can the convergence problems be circumvented for these signals?

As we will see, the application of oversampling leads to uniformly convergent reconstruction processes. Oversampling creates a degree of freedom in the choice of the reconstruction kernel [13]. It is not surprising that an appropriate choice of the kernel will give a reconstruction process, which is uniformly convergent on the whole real axis. This is a classical result [14], but, as we will show, an elaborate kernel design is not necessary as far as only convergence is important. Even the Shannon sampling series with slightly increased bandlimit is uniformly convergent on whole of R when oversampling is applied.

Moreover, we analyze the question whether the redundancy in the samples is sufficient for uniform convergence of the reconstruction process. If so, it would be possible to project the reconstruction process onto the frequency range of the signal. The analysis involves sampling series of the Shannon type with bandlimit [pi]. But the answer is no.

There are many topics of signal theory where oversampling is crucial. One example is the estimation of the peak value [[parallel]f[parallel].sub.[infinity]] of a signal f [member of] [B.sup.[infinity].sub.[pi]] by its samples on the lattice k/a, a > 1, k [member of] Z. The best possible estimate is given by

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

It is interesting to note that the expression 1/cos ([pi]/2a) also appears in the analysis of the Shannon sampling series with oversampling.

2 Notation and Signal Spaces

Let [L.sup.p](R), 1 [less than or equal to] p < [infinity], denote the space of all to the pth power Lebesgue integrable signals on R, with the usual norm [[parallel] x [parallel].sub.p] and [L.sup.[infinity]] (R) the space of all signals for which the essential supremum norm [[parallel] x [parallel].sub.[infinity]] is finite. Furthermore, [l.sup.p], 1 [less than or equal to] p < [infinity], is the space of all sequences such that the p-norm [[parallel] x [parallel].sub.p] is finite, and [l.sup.[infinity]] denotes the space of bounded sequences with the supremum norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [??] denote the Fourier transform of a signal f, where [??] is to be understood in the distributional sense. For [sigma] > 0 let [B.sub.[sigma]] be the set of all entire functions f with the property that for all [epsilon] > 0 there exists a constant C([epsilon]) with [absolute value of f(z)] [less than or equal to] C(epsilon]) exp(([sigma] + [epsilon])[absolute value of z]) for all z [member of] C. The Bernstein space [B.sup.p.sub.[sigma]] consists of all signals in [B.sub.[sigma]], whose restriction to the real line is in [L.sup.p](R), 1 [less than or equal to] p [less than or equal to] [infinity]. A signal in [B.sup.p.sub.[sigma]] is called bandlimited to [sigma]. By the Paley-Wiener-Schwartz theorem, the Fourier transform of a signal bandlimited to a is supported in [-[sigma], [sigma]]. For 1 [less than or equal to] p [less than or equal to] 2 the Fourier transformation is defined in the classical and for p > 2 in the distributional sense. It is well known, that [B.sup.p.sub.[sigma]] [subset] [B.sup.s.sub.[sigma]] for 1 [less than or equal to] p [less than or equal to] s [less than or equal to] [infinity]. Hence, every signal f [member of] [B.sup.p.sub.[sigma]], 1 [less than or equal to] p [less than or equal to] [infinity], is bounded.

At this point, it is interesting to note, that there are areas where no oversampling is necessary: for example, the characterization of f [member of] [B.sup.p.sub.[pi]], 1 < p < [infinity] by means of its samples in the Plancherel-Polya inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [C.sub.1](p) and [C.sub.2](p) are two constants, which depend only on p.

For [sigma] > 0 and 1 [less than or equal to] p [less than or equal to] [infinity] we denote by [PW.sup.p.sub.[sigma]] the Paley-Wiener space of signals f with a representation f(z) = 1/2[pi] [[integral].sup.[sigma].sub.-[sigma]] g([omega])[e.sup.iz[omega]] d[omega], z [member of] C, for some g [member of] [L.sup.p][-[sigma], [sigma]]. If f [member of] [PW.sup.p.sub.[sigma]] then g(w) = [??](w). The norm for [PW.sup.p.sub.[sigma]], [less than or equal to] p [less than or equal to] [infinity], is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As a consequence of Parseval's equality we have [B.sup.2.sub.[sigma] = [PW.sup.2.sub.[sigma]]. Furthermore, the Hausdorff-Young inequality leads to [B.sup.q.sub.[sigma]] [contains] [PW.sup.p.sub.[sigma]] for 1 < p [less than or equal to] 2, 1/p + 1/q = 1 and Holder's inequality to [PW.sup.p.sub.[sigma]] [contains] [PW.sup.s.sub.[sigma]] for 1 [less than or equal to] p < s. Moreover, it holds [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since it is desirable to have a uniformly convergent reconstruction for as large a space of signals as possible, we focus our analysis on the space [PW.sup.1.sub.[pi]], because [PW.sup.1.sub.[pi]] is the largest space in the scale of Paley-Wiener spaces with bandlimit [pi].

The nomenclature concerning the Bernstein and Paley-Wiener spaces that we have introduced so far is not consistent in the literature. Sometimes the space that we call Bernstein space is called Paley-Wiener space [25]. We adhere to the notation used in [16] by Higgins.

The Banach algebra W is the set of continuous signals g with the property that [??] [member of] [L.sup.1](R) exists in the distributional sense and g(t) [[integral].sup.[infinity].sub.-[infinity]] [??]([omega])[e.sup.i[omega]t] d[omega]. The norm is given by [[parallel]f[parallel].sub.w] = 1/2[pi] [[parallel][??][parallel].sub.1] and obviously it holds that [[parallel]f[parallel].sub.[infinity]] [less than or equal to] [[parallel]f[parallel].sub.W]. Note, W is the continuous analog of the Wiener algebra.

Definition 1. Let f be a continuous and bounded signal and [w.sub.g] > 0 fixed. If there is a signal [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called a bandlimited interpolation.

The bandlimited interpolation is a frequently used concept in signal processing [22, p. 144] and it does not need to exist for general continuous and bounded signals. The Banach algebra W is--to the best of our knowledge--the largest known space, all signals of which have a bounded, bandlimited interpolation.

It is interesting that the bandlimited interpolation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exists for all f [member of] W and all [w.sub.g] > 0. The Fourier transform of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where the series in (2) converges for almost all [omega] [member of] [-[[omega].sub.g], [[omega].sub.g]]. Note, since [??] [member of] [L.sup.1] (IR), it holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Furthermore, a simple calculation reveals that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is indeed the Fourier transform of the bandlimited interpolation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The result that all signals in W have a bandlimited interpolation, which is in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], was already obtained by Brown in [6].

Furthermore, for f [member of] W it holds [6] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as well as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Equation (3) shows that every signal f [member of] W can be arbitrarily well approximated by a bandlimited signal [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. However, the construction of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as inverse Fourier integral of (2) is rather impractical. A construction by the use of an sampling series would be convenient. We have for all T > 0 fixed, f [member of] W

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

i.e., the Shannon sampling series converges locally uniformly to the bandlimited interpolation (compare Theorem 1).

It would be of practical relevance to have the uniform convergence in equation (4) on whole R. Together with equation (3) it would then be possible to bound the approximation error, which is made by the finite Shannon sampling series, on the whole real axis. For example, if f [member of] W is a continuous-time signal, then the bandlimit [[omega].sub.g] and the approximation order N, i.e., the order of the partial sum, could be specified, such that for a given [epsilon] > 0 the approximation error is less than c for every point in time.

3 Signal Reconstruction without Oversampling

In the following section the behavior of sampling series without oversampling is analyzed.

First, consider the Shannon sampling series with equidistant samples:

[[infinity].summation over (k=-[infinity]] f(k) sin([pi](t - k))/[pi] (t - k).

A well-known fact [6, 9, 10] is expressed by the following theorem.

Theorem 1. For all f [member of] [PW.sup.1.sub.[pi]] and T > 0 fixed it holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In other words, the symmetric Shannon sampling series is uniformly convergent on compact subsets of R for all f [member of] [PW.sup.1.sub.[pi]].

Although it would be very desirable, Theorem 1 cannot be extended to uniform convergence on whole of R. This behavior could be expected for the Shannon sampling series for the following reason: Since f [member of] [PW.sup.1.sub.[pi]], we have [lim.sub.[absolute value of t][right arrow][infinity]] f(t) = 0 by the Riemann-Lebesgue lemma. Furthermore, the finite Shannon sampling series vanishes for infinite t, too. Thus, both the signal and the partial sums are zero for infinite t. Together with the good (uniform) local convergence behavior, one could conjecture the same good (uniform) convergence behavior on the whole real axis. However, this is not the case.

As it was recently shown for a very general class of reconstruction processes, it is not possible to have a reconstruction based on the samples of the signal, taken uniformly at Nyquist rate, that is uniformly convergent on compact subsets of R and uniformly bounded on whole of R for all signals f [member of] [PW.sup.1.sub.[pi]] [2].

However, for f [member of] [PW.sup.2.sub.[pi]] the situation is different. Throughout the paper we assume that the sequence of sampling points [{[t.sub.k]}.sub.k[member of]Z] is a sequence of strictly increasing real numbers, i.e., [t.sub.k] [member of] R, [t.sub.k] < [t.sub.k+1] for all k [member of] Z. We begin our discussion with the biorthogonal system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with [h.sub.k](t) = (sin [pi](t - [t.sub.k]))/([pi](t - [t.sub.k])), k [member of] Z, which, by definition, has the property

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Furthermore, we assume that [{[[pi].sub.k]}.sub.k[member of]Z] forms a Riesz basis for [PW.sup.2.sub.[pi]], i.e., there are two constants A, B > 0 such that for all M, N [greater than or equal to] 0 and all finite complex numbers [c.sub.l], - M [less than or equal to] l [less than or equal to] N one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

This condition is equivalent to the condition that the sequence [{[t.sub.k]}.sub.k[member of]Z] is a complete interpolating sequence for [PW.sup.2.sub.[pi]] and coefficient space [l.sup.2].

Definition 2. We say that [{[t.sub.k]}.sub.k[member of]Z] is a complete interpolating sequence for [PW.sup.2.sub.[pi]] and coefficient space [l.sup.2] if the interpolation problem f([t.sub.k]) = [c.sub.k], k [member of] Z has exactly one solution f [member of] [PW.sup.2.sub.[pi]] whenever [{[c.sub.k]}.sub.k[member of]Z] [member of] [l.sup.2].

By virtue of equation (5) one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all signals f [member of] [PW.sup.2.sub.[pi]].

Next, the behavior of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

is analyzed for f [member of] [PW.sup.1.sub.[pi]]. There is an entire function [phi] of exponential type [pi] with [phi]([t.sub.k]) = 0, k [member of] Z, and for [[phi].sub.k] we have

[[phi].sub.k](t) = [phi](t)/[phi]'([t.sub.k])(t - [t.sub.k]), k [member of] Z. (7)

Since [phi] has only zeros of order 1, it can be easily seen that [phi]'([t.sub.k]) = [(-1).sup.k] [absolute value of [phi]'([t.sub.k])], k [member of] Z if [phi]'([t.sub.0]) > 0.

Furthermore, if the sequence of sampling points [{[t.sub.k]}.sub.k[member of]Z], [t.sub.k] [member of] R, k [member of] Z is a complete interpolating sequence for [PW.sup.2.sub.[pi]], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

converges uniformly on [absolute value of z] < R for all R < [infinity] and [phi] is an entire function of exponential type [pi] [19].

For [bar.[delta]] < 1/4 the sequence [{[t.sub.k]}.sub.k[member of]Z] is a complete interpolating sequence for [PW.sup.2.sub.[pi]] if [absolute value of [t.sub.k] - k] < [bar.[delta]] and we have the following known facts [28]:

1. There exists a constant [C.sub.1] such that for every k [member of] Z there is a function [[phi].sub.k] [member of] [PW.sup.2.sub.[pi]] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. There are two constants [C.sub.2], [C.sub.3] > 0 such that for all f [member of] [PW.sup.2.sub.[pi]]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 1. A simple example illustrating the aforementioned properties is [phi](t) = sin([pi]t) with [t.sub.k] = k, k [member of] Z. Then [phi]'([t.sub.k]) = [pi] cos([pi][t.sub.k]) = [pi] [(-1).sup.k] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

First of all we want to analyze the operator norm of the sampling series (6).

Lemma 1. Let [{[t.sub.k]}.sub.k[member of]Z] be a complete interpolating sequence for [PW.sup.2.sub.[pi]] with [absolute value of [t.sub.k] - k] [bar.[delta]] for all k [member of] Z and some [bar.[delta]] < 1/3 and [phi] be defined as in equation (8). Then there exists a constant [C.sub.4] > 0 with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10).

Proof. Since the proof of (10) is simple, we omit it and start with the proof of (9). First, we must look at possible signal values f([t.sub.k]), [absolute value of k] [less than or equal to] N and f [member of] [PW.sup.1.sub.[pi]]. Let M > 1 be an arbitrary natural number and consider the signal [g.sub.M] [member of] [PW.sup.1.sub.[pi]] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

after using the identity sin([pi](t - k)) = sin([pi]t)[(-1).sup.k] for t [member of] R, k [member of] Z.

For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the distance [absolute value of f(k)-f([t.sub.k])] can be upper bounded by using

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [t.sub.k] = k [[delta].sub.k], [[delta].sub.k] [less than or equal to] [bar.[delta]] and consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now let N [member of] N be arbitrary but fixed. Obviously, for every [epsilon] > 0 there is a [M.sub.0] = [M.sub.0]([epsilon]), such that

[absolute value of [(-1).sup.k][g.sub.M](k) - 1] < [epsilon]

for all [absolute value of k] [less than or equal to] N and all M [greater than or equal to] [M.sub.0]. Therefore,

[absolute value of [g.sub.M](k) - [(-1).sup.k]] < [epsilon] for all [absolute value of k] [less than or equal to] N and all M [greater than or equal to] [M.sub.0]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, [??] is chosen to satisfy [??] < (1 + [bar.C])/2. Then,

[absolute value of [(-1).sup.k] - [g.sub.M]([t.sub.k])] < 1 + [bar.C]/2 < 1

and

[g.sub.M]([t.sub.k]) = [(-1).sup.k]absolute value of [g.sub.M]([t.sub.k])] (11)

holds for all [absolute value of k] < N and M [greater than or equal to] [M.sub.0]([??]). Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

it follows that

[absolute value of [g.sub.M]([t.sub.k])] > 1 - 1 + [bar.C]/2 = 1 - [bar.C]/2 (12)

for all [absolute value of k] [less than or equal to] N and M [greater than or equal to] [M.sub.0]([??]). For t > [t.sub.N] we get, using equation (11) and

(12),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The case t < [t.sub.-N] is treated analogously.

Remark 1. In general it is not possible to find a constant [C.sub.4] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, it is not possible to replace the maximum in (9) with the maximum over whole R, i.e., the restriction to t [member of] [??]\[[t.sub.-N],[t.sub.N]] is essential. The following example will illustrate this.

Example 2. Consider the Shannon sampling series

([S.sub.N]f)(t) = [N.summation of (k=-N)] f(k) sin([pi](t - k))/[pi](t - k)

with t [member of] [??] fixed. Then Theorem 1 is equivalent to the following result: For T > 0, fixed, there is a constant [C.sub.1] (T) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

holds for all t [member of] [-T, T]. Obviously, one has no connection between the left-hand side of equation (13) and

[N.summation of (k=-N)] [absolute value of sin([pi](t - k))/[pi](t - k)] (14)

in the sense that (14) is a lower bound for the left-hand side of equation (13), because there exists a constant [C.sub.5] such that

[N.summation of (k=-N)] [absolute value of sin([pi](t - k))/[pi](t - k)] [greater than or equal to] [C.sub.5] [absolute value of sin ([pi]t)] log N.

Thus, (14) is only an upper bound for (13).

This result shows that interpolation tasks are relatively difficult to solve in the Paley-Wiener space [PW.sup.1.sub.[pi]]. There is, as it has been shown in Lemma 1, a constant [C.sub.1] such that for each N [member of] N a signal [f.sub.N] [member of] [PW.sup.1.sub.[pi]] can be found with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the property [f.sub.N](k) = [(-1).sup.k] for [absolute value of k] [less than or equal to] N. Furthermore, because of Theorem 1 and the bound for (14), a constant [C.sub.6] must exist such that for all g [member of] [PW.sup.1.sub.[pi]] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we always have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 2. The difference in the behavior of [f.sub.N] and g is caused only by a change of the leading sign. A change of the leading sign is not untypical in practice: The causal part of the signal [f.sub.N] is for example given by ([f.sub.N](k) - g(k))/2.

Lemma 1 can be used to analyze the behavior of (6).

Theorem 2. Let {[t.sub.k]}.sub.k[member of]Z] be defined as in Lemma 1 and [phi] as in equation (8). Furthermore, let [phi] [member of] [B.sup.[infinity].sub.[pi]]. If there is a constant [C.sub.7] > 0 such that for all N [member of] N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

then there exists a constant [C.sub.8] > 0 such that

[parallel][S.sub.N][parallel] > [C.sub.8] log (3/4 N + 1).

Particularly, there is a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. We start with the result from Lemma 1 and obtain for t [member of] ([t.sub.N], [t.sub.N+1])

that

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

Note that for [phi] [member of] [B.sup.[infinity].sub.[phi]] the Bernstein inequality implies that [[parallel][phi]'[parallel].sub.[infinity] [less than or equal to] [pi] [[parallel][phi][parallel].sub.[infinity]], i.e., [absolute value of [phi]'([t.sub.k])] [less than or equal to] [pi][[parallel][phi][parallel].sub.[infinity]]. On the other hand, by assumption, there is a constant [C.sub.7] and a [t.sub.N] [member of] ([t.sub.N],[t.sub.N+1]) such that [absolute value of [phi]([[??].sub.N])] [greater than or equal to] [C.sub.7]. Thus, it follows from (16) that

[parallel][S.sub.N][parallel] > [C.sub.8] log (3/4 N + 1)

for all M [greater than or equal to] [M.sub.0]([??]). Applying the Banach-Steinhaus theorem completes the proof.

The following corollary is a simple consequence of Theorem 2.

Corollary 1. Let {[t.sub.k]}.sub.k[member of]Z] be defined as in Lemma 1 and [phi] as in equation (8). Furthermore, let [phi] [member of] [B.sup.[infinity].sub.[phi]]. Then there exists a constant [C.sub.8] > 0 and a signal [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The next theorem gives a divergence result equal to that in Theorem 2. However, the preconditions are different and no statement about the divergence speed is made in Theorem 3. In Theorem 2 we need the additional condition (15), which is of course restrictive. It is interesting to analyze the situation if the the lower and upper bound from Lemma 1 can be coupled.

Theorem 3. Let {[t.sub.k]}.sub.k[member of]Z] be defined as in Lemma 1 and [phi] as in equation (8). Furthermore, let [phi] [member of] [B.sup.[infinity].sub.[phi]]. If there is a constant [C.sub.10] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

for all N [member of] N, then there exists a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

Remark 3. Note that the requirement (17) is, for example, fulfilled for [phi](t) = sin([pi]t).

Proof. From equation (17) and (9) we know that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assume that there is no signal f [member of] [PW.sup.1.sub.[pi]] which satisfies equation (18). Then by the Banach-Steinhaus theorem there is a constant C12 such that

[parallel][S.sub.N][parallel] [less than or equal to] [C.sub.12] (19)

for all N [member of] N. Therefore,

[N.summation over (k=-N)][absolute value of [phi](t)]/[absolute value of [phi]'([t.sub.k])] [absolute value of t - [t.sub.k]] [less than or equal to] [C.sub.13]

holds for all N [member of] N, t [member of] R and

[[infinity].summation over (k=-[infinity])][absolute value of [phi](t)]/[absolute value of [phi]'([t.sub.k])] [absolute value of t - [t.sub.k]] [less than or equal to] [C.sub.13] (20)

all t [member of] R, because [C.sub.13] is independent of N. Let [epsilon] > 0 be arbitrary and consider the function [phi]((1 - [epsilon])t), t [member of] R, which is obviously in [B.sup.[infinity].sub.(1-[epsilon])[pi]]. Moreover,

[g.sub.[epsilon]](t) = [phi]((1 - [epsilon])t) sin([pi][epsilon]t)/[phi][epsilon]t, t [member of] R,

is a signal in [PW.sup.2.sub.[pi]] and consequently

[g.sub.[epsilon]](t) = [[infinity].summation over (k=-[infinity])] [g.sub.[epsilon]](t)(t.sub.k][[phi].sub.k](t)

for all t [member of] R. Let t [member of] R, t [not equal to] [t.sub.k], k [member of] Z arbitrary but fixed and [delta] > 0 arbitrary. Due to equation (20) there is a [k.sub.0] = [k.sub.0]([delta]) such that

[summation over ([absolute value of k)[greater than or equal to][k.sub.0])] [absolute value of [[phi].sub.k](t)][less than or equal to] [delta].

This leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Inequality (21) is valid for all [epsilon] > 0, but the right-hand side of (21) is independent of [epsilon]. There is a [[epsilon].sub.0] = [[epsilon].sub.0]([delta]) such that

[absolute value of [phi](t) - [g.sub.[epsilon]](t)] < [delta]

and

[absolute value of [phi]([t.sub.k]) - [g.sub.[epsilon]]([t.sub.k])] = [g.sub.[epsilon]]([t.sub.k]) < [delta], [absolute value of k] [less than or equal to] [k.sub.0]

holds for all [epsilon] [less than or equal to] [[epsilon].sub.0]. Consequently, for all 0 < [epsilon] [less than or equal to] [[epsilon].sub.0]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [delta] > 0 is arbitrary, it follows that [phi](t) = 0. But we assumed t [not equal to] [t.sub.k], k [member of] Z, which leads to a contradiction. Thus, (19) cannot hold and (18) must be true.

The Kadec 1/4-Theorem [28, p. 42] answers the question when the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] forms a Riesz basis for [L.sup.2][-[pi], [pi]].

Theorem 4 (Kadec's 1/4-Threorem). If {[t.sub.k]}.sub.k[member of][Z]. is a sequence of real numbers for which

[absolute value of [t.sub.k] - k] [less than or equal to] [bar.[delta]] < 1/4, k [member of Z,

then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] forms a Riesz basis for [L.sup.2][-[pi], [pi]].

It is remarkable that for [bar].[delta]] < 1/4 the discussed reconstruction process is uniformly convergent on R for signals in [PW.sup.2.sub.[pi]], and for signals in [PW.sup.1.sub.[pi]] not even uniformly bounded if the functions [phi] fulfill certain conditions. Furthermore, the considered set of sampling points is minimal in the sense that the removal of one point eliminates the uniqueness in the interpolation problem. We will see in Section 4 that, as soon as the assumption of minimality is dropped, a globally, uniformly convergent reconstruction is possible even for [PW.sup.1.sub.[pi]].

In this section we discussed sampling series of the shape

[N.summation over (k=-N)] f([t.sub.k])[[phi].sub.k](t), t [member of] R, (22)

where the points [{[t.sub.k]}.sub.k[member of]Z] formed a complete interpolating sequence for the space [PW.sup.2.sub.[pi]]. It was shown that for signals f [member of] [PW.sup.1.sub.[pi]] the sampling series are neither globally uniformly convergent nor uniformly bounded on R, in general. In order to show this, it was important to require minimality and to pose restrictions on the bandwidth of the reconstruction process. In the next section it is shown that a global uniform convergence is possible if minimality and minimal bandwidth are waived. However, it is not clear whether oversampling and a larger bandwidth are really necessary in order to achieve a locally uniformly convergent and globally uniformly bounded reconstruction of all signals f [member of] [PW.sup.1.sub.[pi]]. We suppose that this is indeed necessary, i.e., our conjecture is as follows.

Conjecture 1. For every complete interpolating sequence [{[t.sub.k]}.sub.k[member of]Z] for [PW.sup.2.sub.[pi]] there exists a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [phi] defined as in equation (8).

Even if (22) is not uniformly convergent on whole R, it might be uniformly convergent on compact subsets of R. If this is true does not only depend on f but also on the properties of the functions [[phi].sub.k], k [member of] Z. For [[phi].sub.k](t) = sin([pi] (t - k))/([pi](t-k)) and [t.sub.k] = k, k [member of] Z we know from Theorem 1 that (22) is uniformly convergent on compact subsets of R for all f [member of] [PW.sup.1.sub.[pi]]. However, we think that there are Riesz bases where (22) does not converge uniformly on compact subsets of R to f.

Conjecture 2. For all T > 0 there exists a complete interpolating sequence [{[t.sub.k]}.sub.k[member of]Z] for [PW.sup.2.sub.[pi]] and a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [phi] is defined as in equation (8).

Conjecture 2 seems to be difficult to prove. However, we have been able to obtain a weaker result. In [3] we proved that if [{[t.sub.k]}.sub.k[member of]Z] is a complete interpolating sequence for [PW.sup.2.sub.[pi]], then for all t [member of] R there exists a stable linear time-invariant system [T.sub.1] and a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

This is good evidence that the Conjecture 2 is true.

4 Signal Reconstruction with Oversampling

In this section the stability of reconstruction processes with oversampling is analyzed. Oversampling creates a degree of freedom in the choice of the reconstruction kernel [13]. If the kernel is chosen appropriately, then the reconstruction process is uniformly convergent on the whole real axis [5].

As we will see by Theorem 5, even the Shannon sampling series is uniformly convergent on whole of R when oversampling is applied. In order to prove Theorem 5 we need the results from Lemma 2 and 3. However, for better readability we defer the lemmas and begin with Theorem 5.

Theorem 5. Let a > 1 be fixed. Then we have for all f [member of] [PW.sup.1.sub.[pi]] that

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

Proof. Let f [member of] [PW.sup.1.sub.[pi]] be arbitrary but fixed and [member of] > 0. Then there is a [f.sub.1] [member of] [PW.sup.2.sub.[pi]]. such that [f.sub.1](k) is different from zero only for finitely many k [member of] Z and it holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Obviously, [f.sub.1] [member of] [PW.sup.2.sub.[pi]] [subset] [PW.sup.2.sub.[alpha][pi]]. Therefore, there exists a [N.sub.0] = [N.sub.0]([epsilon]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all N [greater than or equal to] [N.sub.0]. Moreover, for all N [greater than or equal to] [N.sub.0],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

where Lemma 2, applied to [f.sub.1] - f, has been used for the last inequality. Since (24) holds for all [member of] > 0, the proof is complete.

A crucial part in the proof was Lemma 2, which can be used to analyze the influence of the oversampling on the peak value of the partial sum of the Shannon sampling series. It is interesting to note, that the 1/cos([pi](2a)) term from equation (1) reappears as integral part of Lemma 2.

Lemma 2. Let a > 1 be fixed. Then we have for all f [member of] [PW.sup.1.sub.[pi]] and all N [member of] N

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

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where we introduced the abbreviation

[g.sub.N](t, [omega], a) := [N.summation over (k=-N)] [e.sup.i[omega]k/a] sin(a[pi](t - k/a))/a[pi](t - k/a)

and used Lemma 3 for the last inequality.

Lemma 3. For all a > 1, t, [omega] [member of] R, [absolute value of [omega]] [less than or equal to] [pi], N [member of] N it holds that

[absolute value of [g.sub.N](t,[omega],a)] [less than or equal to] 2 (1 + 2/cos([pi]/2a)).

Proof. Only the case t > 0 needs to be analyzed. For t = 0 we have [g.sub.N](0,[omega],a) = 1 and the case t < 0 is treated analogously to the case t > 0. If t [member of] A = {k/a: k [member of] N}, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and therefore [absolute value of [g.sub.N](t,[omega],a)] [less than or equal to] 1. Hence, t > 0, t [not member of] A can be assumed. Let N [member of] N be arbitrary but fixed. Obviously,

[g.sub.N](t,[omega],a) = sin(a[pi]t)/a[pi] [N.summation over (k=-N)] [e.sup.ik([omega]/a+[pi])] [1]/[t - k/a] = sin(a[pi]t)/a[pi] [N.summation over (k=-N)] [c.sub.k] [d.sub.k], (26)

where [c.sub.k] [e.sup.ik([omega]/a+[pi])] and [d.sub.k] = 1/(t- k/a). Let [C.sub.k] = [[summation].sup.k.sub.l=1N][c.sub.l], [absolute value of k] [less than or equal to] N. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We begin with the case t > (N + 1)/a. Summation by parts gives

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

The right-hand side of (27) can be further simplified by evaluating the telescoping series

[N-1.summation over (k=-N)] (1/t - k+1/a - 1/t - k/a) = (1/t - N/a - 1/t + N/a [less than or equal to] a (28)

for t > (N + 1)/a. Combining equations (26), (27), and (28) leads to

[absolute value of [g.sub.N](t,[omega],a)] [less than or equal to] 2/[pi] cos([pi]/2a).

Next, the case t < (N + 1)/a is treated. Let [N.sub.t] be the largest natural number such that [N.sub.t]/a < t. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

The first term on the right-hand side of equation (29) can be upper bounded by

[absolute value of [N.sub.t]-1.summation over (k=-N)] [c.sub.k][d.sub.k]] [less than or equal to] 2a/cos([pi]/2a)

exactly in the same way as before and the second term by

[absolute value of N.summation over (k=[N.sub.t]+2)] [c.sub.k][d.sub.k]] [less than or equal to] 2a/cos([pi]/2a).

Theorem 5 shows that if oversampling is used, we can have global uniform convergence for all f [member of] [PW.sup.1.sub.[pi]]. An interesting question concerns the rate of convergence of the sampling series (1): Given some a > 1, can we find two constants [gamma] = [gamma](f, a) > 0 and C = C(f, a) < [infinity] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every f [member of] [PW.sup.1.sub.[pi]]? Is it even possible to find a [gamma], independently of f? The answer to both questions is given by the next theorem.

Theorem 6. Let a > 1 be fixed. For each arbitrary sequence {[[epsilon].sub.N]} of positive numbers that converges to zero, there exists a f [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. In order to accomplish the proof, we introduce the operator

([R.sub.N] f)(t) := f(t) - [N.summation over (k=-N) f (k/a) sin(a[pi](t - k/a))/a[pi](t - k/a).

Obviously we have

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

Since (30) is valid for all t [member of] R, we can choose t = (N + 1)/a. Then we have

[N.summation over (k=-N)] [e.sup.i[omega]k/a] sin(a[pi](t - k/a))/a[pi](t - k/a) = 0

and consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, the operator [[??].sub.N] [member of] N, defined by

([[??].sub.N]f)(t) := 1/[[epsilon].sub.N]([R.sub.N]f)(t),

is linear and bounded, and we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, by the Banach-Steinhaus theorem there is a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that lim [sup.sub.N[right arrow][infinity]] [[parallel][[??].sub.N][f.sub.1][parallel].sub.[infinity]] = [infinity], which completes the proof.

Theorem 6 shows that the convergence speed of the Shannon sampling series with oversampling for the space [PW.sup.1.sub.[pi]] can be arbitrarily slow and that no convergence rates can be given.

5 Non-Symmetric Sampling Series

Many attempts have been made to proof convergence results that are true for the symmetric sampling series of the non-symmetric case. However, it turned out that the non-symmetric sampling series exhibits a significantly different convergence behavior compared to the symmetric sampling series for certain problems. This is particularly true for the Shannon sampling series and the space [PW.sup.1.sub.[pi]]: It has been shown [1, 4] that there is a signal f [member of] [PW.sup.1.sub.[pi]] such that for all t [member of] R/Z

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which means that the non-symmetric Shannon sampling series does not converge to f for all t [member of] R\Z. Consequently, a theorem stating the uniform convergence of the non-symmetric Shannon sampling series on compact subsets of R, similarly to Theorem 1, which states exactly this for the symmetric Shannon sampling series, of course cannot exist.

The preceding considerations have been made for the case where the samples are taken at Nyquist rate, i.e., where no oversampling is applied. The convergence behavior of the non-symmetric Shannon sampling series changes completely if we use oversampling. A closer look at the proof of Lemma 3 shows that the inequality convergence behavior of the non-symmetric Shannon sampling series changes completely if we use oversampling. A closer look at the proof of Lemma 3 shows that the inequality.

[N.summation over (k=-M)] [e.sup.i[omega]k/a] sin([alpha][pi](t - [k/[alpha]))/[alpha][pi](t - k/[alpha]]) [less than or equal to] 2 (1 + [2/cos([pi]/2[alpha]))

holds for all M, N [member of] N, t [member of] R and [absolute value of [omega]] [less than or equal to] [pi]. Therefore, it is possible to get an analogous result to Lemma 3 for the non-symmetric sampling series. Consequently, a theorem similar to Theorem 5 can be derived for the non-symmetric sampling series:

Theorem 7. Let a > 1 be fixed. Then we have for all f [member of] [PW.sup.1.sub[pi]]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

Proof. Analogously to Theorem 5.

We have seen that for f [member of] [PW.sup.1.sub.[pi]] there are differences in the convergence behavior between the symmetric Shannon sampling series and the non-symmetric Shanon sampling series when no oversampling is applied. However, with oversampling both have the same good convergence behavior, i.e., both are uniformly convergent on whole of R.

6 Convergence Analysis

We return to the discussion begun in Section 3, namely the question of finding a sufficient condition for uniform convergence of the reconstruction process. The results from Section 4 can be used to analyze the convergence behavior of the Shannon sampling series for [PW.sup.1.sub.[pi]] signals without oversampling. It has been proven that, for some f [member of] [PW.sup.1.sub.[pi]], the Shannon sampling series

[[infinity].summation over (k=-[infinity])] f(k) sin([pi](t - k))/[pi](t - k)]

does not converge uniformly on whole of R to f. However, if [??]([omega) fulfills certain integrability conditions in the vicinity of [omega] = [pi], even the non-symmetric Shannon sampling series converges uniformly on whole of R.

Theorem 8. If f [member of] [PW.sup.1.sub.[pi]] has the property that there exists a [delta] > 0 and a p > 1 such that

[[infinity].sub.[pi]-[delta][less than or equal to][absolute value of [omega]] [less than or equal to][pi]] [[absolute value of [??]([omega])].sup.p] d[omega] < [infinity],

then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Two auxiliary bandlimited signals [f.sub.1] and [f.sub.2] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are needed for the proof. Obviously, [f.sub.1] [member of] [PW.sup.1.sub.[pi]-[delta]] and f(t) = [f.sub.1](t) + [f.sub.2](t), t [member of] R. Furthermore, it holds by assumption that

1/2[pi] [[infinity].sup.[pi].sub.-[pi]] [[absolute value of [[??].sub.2]([omega])].sup.p] d[omega] < [infinity].

Therefore, [f.sub.2] [member of] [PW.sup.p.sub.[pi]]. Let [epsilon] > 0 be arbitrarily chosen. Then there exists a [N.sub.0] = [N.sub.0]([epsilon]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all N [less than or equal to] [N.sub.0]. Since [f.sub.1] [member of] [PW.sup.1.sub.[pi]-[delta]], there exists a [N.sub.1] = [N.sub.1]([epsilon]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all N [greater than or equal to] [N.sub.1]. Consequently, for all N [greater than or equal to] max([N.sub.0], [N.sub.1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

Since inequality (32) is valid for all t [member of] R, the proof is completed.

Theorem 8 shows that the known divergence of the Shannon sampling series for signals f [member of] [PW.sup.1.sub.-1[pi]] is only a consequence of the behavior of [??]([omega]) in the vicinity of [omega] = [pi]. If certain integrability conditions are fulfilled in this region, then the divergence does not appear.

7 Oversampling and Kernels

In this section we want to examine whether oversampling is really a universal remedy for circumventing convergence problems. We begin with the following observation: The finite sampling series

1/a[pi] [N.summation over (k=-N) f(k/a) sin(a[pi](t - k/a]))/t - k/a (33)

with oversampling factor a > 1 is not bandlimited to [pi], but rather to a[pi]. Can we use functions in the approximation formula that are bandlimited to [pi] itself?

For f [member of] [PW.sup.2.sub.[pi]] this is obviously possible. By expanding [??] into a Fourier series in the interval [-a[pi], a[pi]], a > 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, using the definition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of the characteristic function [[chi].sub.[pi]], it holds that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where we used the Cauchy-Schwarz inequality. Therefore,

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

As a consequence, for f [member of] [PW.sup.2.sub.[pi]] it is possible to use

1/a[pi] [N.summation over (k=-N)] f(k/a) sin([pi](t - k/a))/t - k/a (35)

for signal reconstruction. This has the advantage that f [member of [PW.sup.2.sub.[pi]] can be approximated according to equation (35) by a signal, which is bandlimited to [pi]. The sampling series (35) can be obtained by low-pass filtering the signal generated by the sampling series in equation (33) with bandlimit [pi].

Since for all signals f [member of] [PW.sup.1.sub.[pi]], the series in (33) converges uniformly on whole of R to the signal f, it is reasonable to ask whether a low-pass filtering of (33) preserves the uniform convergence, even for f [member of] [PW.sup.1.sub.[pi]]. Then, this would be the projection on the desired frequency interval. However, the following theorem gives a negative answer.

Theorem 9. There is a [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

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

or equivalently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. For t [member of] R, a > 1, N [member of] N fixed, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next,

[q.sub.N](t, [omega], a) := [N.summation over (k=-N)] [e.sup.i[omega]k/a] sin([pi](t - k/a))/a[pi](t - k/a)

is analyzed. For [omega] = [pi] and [t.sub.N] = (N + 1/2)/a we get

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

The second sum in equation (37) was evaluated in the same way as the sum in (26). Consequently, for [t.sub.N] = (N + 1/2)/a we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, by the Banach-Steinhaus theorem there is a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] that fulfills (36).

The result of Theorem 9 is interesting: it shows that a low-pass filtering destroys the uniform convergence. Not even the uniform boundedness is preserved. This means that the sole redundancy in the set [{f(k/a)}.sub.k[member of]Z], a > 1, is not sufficient for uniform convergence. It is necessary to use a proper kernel. Consequently, a simultaneous projection into the space [PW.sup.1.sub.[pi]] during reconstruction is not possible without the reconstruction process to become divergent.

It is easy to see that for f [member of] [PW.sup.1.sub.[pi]] no reconstruction process of the shape

[N.summation over (k=-N)] f (k/a) [phi] (t - k/a) (38)

and with the property that (38) is bandlimited to [pi] can be uniformly bounded. If (38) is bandlimited to [pi], then the Fourier transform

1/a [N.summation over (k=-N)] f(k/a) [e.sup.i[omega]k/a](a[??]([omega]))

is supported in [[pi], [pi]]. Since [[summation].sup.N.sub.k=-N] f(k/a)[e.sup.iwk/a] has only isolated zeros, [??]([omega]) = 0 for [absolute value of [omega]] > [pi] must hold. Moreover, the assumed convergence of (38) implies that a[??]([omega]) = 1 for [absolute value of [omega]] < [pi]. Hence, [phi](t) = (sin([pi]t))/(a[pi]t). But Theorem 9 has shown that the reconstruction process for this kernel is neither uniformly convergent nor uniformly bounded on whole of R.

We have seen that it is impossible to have a locally uniformly convergent and globally, uniformly bounded reconstruction for all f [member of] [PW.sup.1.sub.[pi]] on the basis of the samples [{f(k/a)}.sub.k[member of]Z] if the reconstruction process is of the shape (38) and [phi] is bandlimited to [pi]. It is possible to approach the bandlimit [pi] arbitrarily closely, but to have [phi] exactly bandlimited with [pi] is impossible if uniform boundedness is desired.

The perception that oversampling is useful for almost everything is prevalent. Indeed, for all signals f [member of] [PW.sup.1.sub.[pi]] a uniformly convergent reconstruction is possible if the signal is known on an oversampling set {f(k/a)}, a > 1. Consider [omega] with a[??]([omega]) = 1 for [absolute value of [omega]] [less than or equal to] [pi], [??]([omega]) = 0 for [absolute value of [omega]] [greater than or equal to] a[pi] and [phi] [member of] [B.sup.1.sub.a[pi]]. Then we have for all f [member of] [PW.sup.1.sub.[pi]]

f(t) = [[infinity].summation over (k=-[infinity]) f(k/a) [phi] (t - k/a),

and since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the sampling series is absolutely convergent.

ACKNOWLEDGEMENT

The authors would like to thank J. Hagenauer for discussions on earlier versions of this paper.

This work was partly supported by the German Research Foundation (DFG) under grant BO 1734/9-1.

The material in this paper was presented in part at the 7th International ITG Conference on Source and Channel Coding (SCC'08), Ulm, Germany, January 2008.

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[24] H. Raabe, Untersuchungen an der wechselzeitigen Mehrfachubertragung (Multiplexubertragung), Elektrische Nachrichtentechnik 16 (1939), 213-228.

[25] K. Seip, Developments from nonharmonic Fourier series, Documenta Mathematica, Proc. ICM, vol. II, 1998, pp. 713-722.

[26] C. E. Shannon, Communication in the presence of noise, Proceedings of the IRE, vol. 37, January 1949, pp. 10-21.

[27] E. T. Whittaker, On the functions which are represented by the expansions of the interpolation theory, Proceedings of the Royal Society of Edinburgh 35 (1915), 181-194.

[28] R. M. Young, An introduction to nonharmonic Fourier series, Academic Press, October 1980.

[29] M. Zakai, Band-limited functions and the sampling theorem, Information and Control 8 (1965), no. 2, 143-158.

Holger Boche

Heinrich-Hertz-Chair for Mobile Communications

Technische Universitat Berlin

Einsteinufer 25

D-10578 Berlin, Germany

holger.boche@mk.tu-berlin.de

Ullrich J. Monich

Heinrich-Hertz-Chair for Mobile Communications

Technische Universitat Berlin

Einsteinufer 25

D-10578 Berlin, Germany

ullrich.moenich@mk.tu-berlin.de

(1) This question was asked by one of the reviewers.

For signals in the Paley-Wiener space [PW.sup.1.sub.[pi]] a reconstruction in the form of a sampling series that is uniformly convergent on compact subsets of the real axis and uniformly bounded on the whole real axis is not possible in general if the signals are sampled equidistantly at Nyquist rate. We prove that, even if the signal is non-uniformly sampled with an average sampling rate equal to the Nyquist rate, a uniformly convergent reconstruction is not possible. Additionally, we provide a detailed convergence analysis and give a sufficient condition for the uniform convergence of the Shannon sampling series without oversampling. However, if oversampling is applied, a uniformly convergent reconstruction is always possible and as far as convergence is concerned no elaborate kernel design is necessary. Moreover, we show that a projection of the reconstruction process onto the range of signal frequencies is not possible without losing the good convergence behavior.

Key words and phrases : Sampling Theorem, Oversampling, Paley-Wiener Space, Signal Reconstruction, Uniform Convergence, Complete Interpolating Sequence, Riesz Basis

2000 AMS Mathematics Subject Classification 94A20, 94A05

1 Introduction

A reconstruction of signals from its samples by means of a sampling series that is uniformly convergent on compact subsets and uniformly bounded on the whole real axis is important not only from the theoretical point of view but in particular for practical applications. For the Paley-Wiener space [PW.sup.2.sub.[pi]] there is one certain sampling series that has become an integral part of modern communication and information theory. This sampling series is called Shannon sampling series, nowadays, although it has also been found by others, either earlier or independently at the same time. Whittaker introduced it in the mathematical literature in 1915 [27]. Several years later the sampling theorem appeared in the engineering literature: Kotel'nikov published it in 1933 [18], Raabe in 1939 [24] and Shannon in 1949 [26]. Since the work by Kotel'nikov was published in Russian and Shannon's paper was already written in 1940 [10], it is reasonable to assume that all three publications were created independently. See [10] and [20] for further historical details and [21, Chapter 2] for an overview of the topic. Recent developments in sampling theory can be found in [15, 14]. Chapter 3 of [21] deals with non-uniform sampling.

However, for practical applications the analysis of the space [PW.sup.2.sub.[pi]] is often not sufficient. Therefore, extensions in several directions have been considered, e.g., sampling theorems for more general function spaces [29, 11, 23, 17], sampling series with more general kernels (bandlimited and non-bandlimited) [14, Chapter 6] [7, 8, 12], multiband sampling and multidimensional sampling [14].

In this paper we do the analysis for the space [PW.sup.1.sub.[pi]], which is the largest space in the scale of Paley-Wiener spaces, because this space has some remarkable properties, e.g., the existence of the bandlimited interpolation. Therefore, this paper can be considered to be a continuation of the classical work by Brown and Butzer.

For signals f [member of] [PW.sup.1.sub.[pi]] there exists no uniformly convergent reconstruction process that uses only samples taken equidistantly at Nyquist rate, although this set of samples constitutes a set of uniqueness. It has been shown that this result is valid not only for the Shannon sampling series, but also for a fairly general class of reconstruction processes [2]. In this paper we consider non-equidistant sampling, with the sampling points [{[t.sub.k]}.sub.k[member of]Z] being real and satisfying [absolute value of [t.sub.k]-k] [less than or equal to] [delta] < 1/3, [member of] Z. We show that for a whole class of sampling series there are signals f [member] [PW.sup.1.sub.[pi]] such that neither global uniform convergence nor global boundedness is given. The question is: how can the convergence problems be circumvented for these signals?

As we will see, the application of oversampling leads to uniformly convergent reconstruction processes. Oversampling creates a degree of freedom in the choice of the reconstruction kernel [13]. It is not surprising that an appropriate choice of the kernel will give a reconstruction process, which is uniformly convergent on the whole real axis. This is a classical result [14], but, as we will show, an elaborate kernel design is not necessary as far as only convergence is important. Even the Shannon sampling series with slightly increased bandlimit is uniformly convergent on whole of R when oversampling is applied.

Moreover, we analyze the question whether the redundancy in the samples is sufficient for uniform convergence of the reconstruction process. If so, it would be possible to project the reconstruction process onto the frequency range of the signal. The analysis involves sampling series of the Shannon type with bandlimit [pi]. But the answer is no.

There are many topics of signal theory where oversampling is crucial. One example is the estimation of the peak value [[parallel]f[parallel].sub.[infinity]] of a signal f [member of] [B.sup.[infinity].sub.[pi]] by its samples on the lattice k/a, a > 1, k [member of] Z. The best possible estimate is given by

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

It is interesting to note that the expression 1/cos ([pi]/2a) also appears in the analysis of the Shannon sampling series with oversampling.

2 Notation and Signal Spaces

Let [L.sup.p](R), 1 [less than or equal to] p < [infinity], denote the space of all to the pth power Lebesgue integrable signals on R, with the usual norm [[parallel] x [parallel].sub.p] and [L.sup.[infinity]] (R) the space of all signals for which the essential supremum norm [[parallel] x [parallel].sub.[infinity]] is finite. Furthermore, [l.sup.p], 1 [less than or equal to] p < [infinity], is the space of all sequences such that the p-norm [[parallel] x [parallel].sub.p] is finite, and [l.sup.[infinity]] denotes the space of bounded sequences with the supremum norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [??] denote the Fourier transform of a signal f, where [??] is to be understood in the distributional sense. For [sigma] > 0 let [B.sub.[sigma]] be the set of all entire functions f with the property that for all [epsilon] > 0 there exists a constant C([epsilon]) with [absolute value of f(z)] [less than or equal to] C(epsilon]) exp(([sigma] + [epsilon])[absolute value of z]) for all z [member of] C. The Bernstein space [B.sup.p.sub.[sigma]] consists of all signals in [B.sub.[sigma]], whose restriction to the real line is in [L.sup.p](R), 1 [less than or equal to] p [less than or equal to] [infinity]. A signal in [B.sup.p.sub.[sigma]] is called bandlimited to [sigma]. By the Paley-Wiener-Schwartz theorem, the Fourier transform of a signal bandlimited to a is supported in [-[sigma], [sigma]]. For 1 [less than or equal to] p [less than or equal to] 2 the Fourier transformation is defined in the classical and for p > 2 in the distributional sense. It is well known, that [B.sup.p.sub.[sigma]] [subset] [B.sup.s.sub.[sigma]] for 1 [less than or equal to] p [less than or equal to] s [less than or equal to] [infinity]. Hence, every signal f [member of] [B.sup.p.sub.[sigma]], 1 [less than or equal to] p [less than or equal to] [infinity], is bounded.

At this point, it is interesting to note, that there are areas where no oversampling is necessary: for example, the characterization of f [member of] [B.sup.p.sub.[pi]], 1 < p < [infinity] by means of its samples in the Plancherel-Polya inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [C.sub.1](p) and [C.sub.2](p) are two constants, which depend only on p.

For [sigma] > 0 and 1 [less than or equal to] p [less than or equal to] [infinity] we denote by [PW.sup.p.sub.[sigma]] the Paley-Wiener space of signals f with a representation f(z) = 1/2[pi] [[integral].sup.[sigma].sub.-[sigma]] g([omega])[e.sup.iz[omega]] d[omega], z [member of] C, for some g [member of] [L.sup.p][-[sigma], [sigma]]. If f [member of] [PW.sup.p.sub.[sigma]] then g(w) = [??](w). The norm for [PW.sup.p.sub.[sigma]], [less than or equal to] p [less than or equal to] [infinity], is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As a consequence of Parseval's equality we have [B.sup.2.sub.[sigma] = [PW.sup.2.sub.[sigma]]. Furthermore, the Hausdorff-Young inequality leads to [B.sup.q.sub.[sigma]] [contains] [PW.sup.p.sub.[sigma]] for 1 < p [less than or equal to] 2, 1/p + 1/q = 1 and Holder's inequality to [PW.sup.p.sub.[sigma]] [contains] [PW.sup.s.sub.[sigma]] for 1 [less than or equal to] p < s. Moreover, it holds [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since it is desirable to have a uniformly convergent reconstruction for as large a space of signals as possible, we focus our analysis on the space [PW.sup.1.sub.[pi]], because [PW.sup.1.sub.[pi]] is the largest space in the scale of Paley-Wiener spaces with bandlimit [pi].

The nomenclature concerning the Bernstein and Paley-Wiener spaces that we have introduced so far is not consistent in the literature. Sometimes the space that we call Bernstein space is called Paley-Wiener space [25]. We adhere to the notation used in [16] by Higgins.

The Banach algebra W is the set of continuous signals g with the property that [??] [member of] [L.sup.1](R) exists in the distributional sense and g(t) [[integral].sup.[infinity].sub.-[infinity]] [??]([omega])[e.sup.i[omega]t] d[omega]. The norm is given by [[parallel]f[parallel].sub.w] = 1/2[pi] [[parallel][??][parallel].sub.1] and obviously it holds that [[parallel]f[parallel].sub.[infinity]] [less than or equal to] [[parallel]f[parallel].sub.W]. Note, W is the continuous analog of the Wiener algebra.

Definition 1. Let f be a continuous and bounded signal and [w.sub.g] > 0 fixed. If there is a signal [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called a bandlimited interpolation.

The bandlimited interpolation is a frequently used concept in signal processing [22, p. 144] and it does not need to exist for general continuous and bounded signals. The Banach algebra W is--to the best of our knowledge--the largest known space, all signals of which have a bounded, bandlimited interpolation.

It is interesting that the bandlimited interpolation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exists for all f [member of] W and all [w.sub.g] > 0. The Fourier transform of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where the series in (2) converges for almost all [omega] [member of] [-[[omega].sub.g], [[omega].sub.g]]. Note, since [??] [member of] [L.sup.1] (IR), it holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Furthermore, a simple calculation reveals that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is indeed the Fourier transform of the bandlimited interpolation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The result that all signals in W have a bandlimited interpolation, which is in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], was already obtained by Brown in [6].

Furthermore, for f [member of] W it holds [6] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as well as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Equation (3) shows that every signal f [member of] W can be arbitrarily well approximated by a bandlimited signal [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. However, the construction of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as inverse Fourier integral of (2) is rather impractical. A construction by the use of an sampling series would be convenient. We have for all T > 0 fixed, f [member of] W

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

i.e., the Shannon sampling series converges locally uniformly to the bandlimited interpolation (compare Theorem 1).

It would be of practical relevance to have the uniform convergence in equation (4) on whole R. Together with equation (3) it would then be possible to bound the approximation error, which is made by the finite Shannon sampling series, on the whole real axis. For example, if f [member of] W is a continuous-time signal, then the bandlimit [[omega].sub.g] and the approximation order N, i.e., the order of the partial sum, could be specified, such that for a given [epsilon] > 0 the approximation error is less than c for every point in time.

3 Signal Reconstruction without Oversampling

In the following section the behavior of sampling series without oversampling is analyzed.

First, consider the Shannon sampling series with equidistant samples:

[[infinity].summation over (k=-[infinity]] f(k) sin([pi](t - k))/[pi] (t - k).

A well-known fact [6, 9, 10] is expressed by the following theorem.

Theorem 1. For all f [member of] [PW.sup.1.sub.[pi]] and T > 0 fixed it holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In other words, the symmetric Shannon sampling series is uniformly convergent on compact subsets of R for all f [member of] [PW.sup.1.sub.[pi]].

Although it would be very desirable, Theorem 1 cannot be extended to uniform convergence on whole of R. This behavior could be expected for the Shannon sampling series for the following reason: Since f [member of] [PW.sup.1.sub.[pi]], we have [lim.sub.[absolute value of t][right arrow][infinity]] f(t) = 0 by the Riemann-Lebesgue lemma. Furthermore, the finite Shannon sampling series vanishes for infinite t, too. Thus, both the signal and the partial sums are zero for infinite t. Together with the good (uniform) local convergence behavior, one could conjecture the same good (uniform) convergence behavior on the whole real axis. However, this is not the case.

As it was recently shown for a very general class of reconstruction processes, it is not possible to have a reconstruction based on the samples of the signal, taken uniformly at Nyquist rate, that is uniformly convergent on compact subsets of R and uniformly bounded on whole of R for all signals f [member of] [PW.sup.1.sub.[pi]] [2].

However, for f [member of] [PW.sup.2.sub.[pi]] the situation is different. Throughout the paper we assume that the sequence of sampling points [{[t.sub.k]}.sub.k[member of]Z] is a sequence of strictly increasing real numbers, i.e., [t.sub.k] [member of] R, [t.sub.k] < [t.sub.k+1] for all k [member of] Z. We begin our discussion with the biorthogonal system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with [h.sub.k](t) = (sin [pi](t - [t.sub.k]))/([pi](t - [t.sub.k])), k [member of] Z, which, by definition, has the property

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Furthermore, we assume that [{[[pi].sub.k]}.sub.k[member of]Z] forms a Riesz basis for [PW.sup.2.sub.[pi]], i.e., there are two constants A, B > 0 such that for all M, N [greater than or equal to] 0 and all finite complex numbers [c.sub.l], - M [less than or equal to] l [less than or equal to] N one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

This condition is equivalent to the condition that the sequence [{[t.sub.k]}.sub.k[member of]Z] is a complete interpolating sequence for [PW.sup.2.sub.[pi]] and coefficient space [l.sup.2].

Definition 2. We say that [{[t.sub.k]}.sub.k[member of]Z] is a complete interpolating sequence for [PW.sup.2.sub.[pi]] and coefficient space [l.sup.2] if the interpolation problem f([t.sub.k]) = [c.sub.k], k [member of] Z has exactly one solution f [member of] [PW.sup.2.sub.[pi]] whenever [{[c.sub.k]}.sub.k[member of]Z] [member of] [l.sup.2].

By virtue of equation (5) one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all signals f [member of] [PW.sup.2.sub.[pi]].

Next, the behavior of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

is analyzed for f [member of] [PW.sup.1.sub.[pi]]. There is an entire function [phi] of exponential type [pi] with [phi]([t.sub.k]) = 0, k [member of] Z, and for [[phi].sub.k] we have

[[phi].sub.k](t) = [phi](t)/[phi]'([t.sub.k])(t - [t.sub.k]), k [member of] Z. (7)

Since [phi] has only zeros of order 1, it can be easily seen that [phi]'([t.sub.k]) = [(-1).sup.k] [absolute value of [phi]'([t.sub.k])], k [member of] Z if [phi]'([t.sub.0]) > 0.

Furthermore, if the sequence of sampling points [{[t.sub.k]}.sub.k[member of]Z], [t.sub.k] [member of] R, k [member of] Z is a complete interpolating sequence for [PW.sup.2.sub.[pi]], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

converges uniformly on [absolute value of z] < R for all R < [infinity] and [phi] is an entire function of exponential type [pi] [19].

For [bar.[delta]] < 1/4 the sequence [{[t.sub.k]}.sub.k[member of]Z] is a complete interpolating sequence for [PW.sup.2.sub.[pi]] if [absolute value of [t.sub.k] - k] < [bar.[delta]] and we have the following known facts [28]:

1. There exists a constant [C.sub.1] such that for every k [member of] Z there is a function [[phi].sub.k] [member of] [PW.sup.2.sub.[pi]] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. There are two constants [C.sub.2], [C.sub.3] > 0 such that for all f [member of] [PW.sup.2.sub.[pi]]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 1. A simple example illustrating the aforementioned properties is [phi](t) = sin([pi]t) with [t.sub.k] = k, k [member of] Z. Then [phi]'([t.sub.k]) = [pi] cos([pi][t.sub.k]) = [pi] [(-1).sup.k] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

First of all we want to analyze the operator norm of the sampling series (6).

Lemma 1. Let [{[t.sub.k]}.sub.k[member of]Z] be a complete interpolating sequence for [PW.sup.2.sub.[pi]] with [absolute value of [t.sub.k] - k] [bar.[delta]] for all k [member of] Z and some [bar.[delta]] < 1/3 and [phi] be defined as in equation (8). Then there exists a constant [C.sub.4] > 0 with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10).

Proof. Since the proof of (10) is simple, we omit it and start with the proof of (9). First, we must look at possible signal values f([t.sub.k]), [absolute value of k] [less than or equal to] N and f [member of] [PW.sup.1.sub.[pi]]. Let M > 1 be an arbitrary natural number and consider the signal [g.sub.M] [member of] [PW.sup.1.sub.[pi]] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

after using the identity sin([pi](t - k)) = sin([pi]t)[(-1).sup.k] for t [member of] R, k [member of] Z.

For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the distance [absolute value of f(k)-f([t.sub.k])] can be upper bounded by using

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [t.sub.k] = k [[delta].sub.k], [[delta].sub.k] [less than or equal to] [bar.[delta]] and consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now let N [member of] N be arbitrary but fixed. Obviously, for every [epsilon] > 0 there is a [M.sub.0] = [M.sub.0]([epsilon]), such that

[absolute value of [(-1).sup.k][g.sub.M](k) - 1] < [epsilon]

for all [absolute value of k] [less than or equal to] N and all M [greater than or equal to] [M.sub.0]. Therefore,

[absolute value of [g.sub.M](k) - [(-1).sup.k]] < [epsilon] for all [absolute value of k] [less than or equal to] N and all M [greater than or equal to] [M.sub.0]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, [??] is chosen to satisfy [??] < (1 + [bar.C])/2. Then,

[absolute value of [(-1).sup.k] - [g.sub.M]([t.sub.k])] < 1 + [bar.C]/2 < 1

and

[g.sub.M]([t.sub.k]) = [(-1).sup.k]absolute value of [g.sub.M]([t.sub.k])] (11)

holds for all [absolute value of k] < N and M [greater than or equal to] [M.sub.0]([??]). Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

it follows that

[absolute value of [g.sub.M]([t.sub.k])] > 1 - 1 + [bar.C]/2 = 1 - [bar.C]/2 (12)

for all [absolute value of k] [less than or equal to] N and M [greater than or equal to] [M.sub.0]([??]). For t > [t.sub.N] we get, using equation (11) and

(12),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The case t < [t.sub.-N] is treated analogously.

Remark 1. In general it is not possible to find a constant [C.sub.4] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, it is not possible to replace the maximum in (9) with the maximum over whole R, i.e., the restriction to t [member of] [??]\[[t.sub.-N],[t.sub.N]] is essential. The following example will illustrate this.

Example 2. Consider the Shannon sampling series

([S.sub.N]f)(t) = [N.summation of (k=-N)] f(k) sin([pi](t - k))/[pi](t - k)

with t [member of] [??] fixed. Then Theorem 1 is equivalent to the following result: For T > 0, fixed, there is a constant [C.sub.1] (T) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

holds for all t [member of] [-T, T]. Obviously, one has no connection between the left-hand side of equation (13) and

[N.summation of (k=-N)] [absolute value of sin([pi](t - k))/[pi](t - k)] (14)

in the sense that (14) is a lower bound for the left-hand side of equation (13), because there exists a constant [C.sub.5] such that

[N.summation of (k=-N)] [absolute value of sin([pi](t - k))/[pi](t - k)] [greater than or equal to] [C.sub.5] [absolute value of sin ([pi]t)] log N.

Thus, (14) is only an upper bound for (13).

This result shows that interpolation tasks are relatively difficult to solve in the Paley-Wiener space [PW.sup.1.sub.[pi]]. There is, as it has been shown in Lemma 1, a constant [C.sub.1] such that for each N [member of] N a signal [f.sub.N] [member of] [PW.sup.1.sub.[pi]] can be found with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the property [f.sub.N](k) = [(-1).sup.k] for [absolute value of k] [less than or equal to] N. Furthermore, because of Theorem 1 and the bound for (14), a constant [C.sub.6] must exist such that for all g [member of] [PW.sup.1.sub.[pi]] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we always have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 2. The difference in the behavior of [f.sub.N] and g is caused only by a change of the leading sign. A change of the leading sign is not untypical in practice: The causal part of the signal [f.sub.N] is for example given by ([f.sub.N](k) - g(k))/2.

Lemma 1 can be used to analyze the behavior of (6).

Theorem 2. Let {[t.sub.k]}.sub.k[member of]Z] be defined as in Lemma 1 and [phi] as in equation (8). Furthermore, let [phi] [member of] [B.sup.[infinity].sub.[pi]]. If there is a constant [C.sub.7] > 0 such that for all N [member of] N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

then there exists a constant [C.sub.8] > 0 such that

[parallel][S.sub.N][parallel] > [C.sub.8] log (3/4 N + 1).

Particularly, there is a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. We start with the result from Lemma 1 and obtain for t [member of] ([t.sub.N], [t.sub.N+1])

that

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

Note that for [phi] [member of] [B.sup.[infinity].sub.[phi]] the Bernstein inequality implies that [[parallel][phi]'[parallel].sub.[infinity] [less than or equal to] [pi] [[parallel][phi][parallel].sub.[infinity]], i.e., [absolute value of [phi]'([t.sub.k])] [less than or equal to] [pi][[parallel][phi][parallel].sub.[infinity]]. On the other hand, by assumption, there is a constant [C.sub.7] and a [t.sub.N] [member of] ([t.sub.N],[t.sub.N+1]) such that [absolute value of [phi]([[??].sub.N])] [greater than or equal to] [C.sub.7]. Thus, it follows from (16) that

[parallel][S.sub.N][parallel] > [C.sub.8] log (3/4 N + 1)

for all M [greater than or equal to] [M.sub.0]([??]). Applying the Banach-Steinhaus theorem completes the proof.

The following corollary is a simple consequence of Theorem 2.

Corollary 1. Let {[t.sub.k]}.sub.k[member of]Z] be defined as in Lemma 1 and [phi] as in equation (8). Furthermore, let [phi] [member of] [B.sup.[infinity].sub.[phi]]. Then there exists a constant [C.sub.8] > 0 and a signal [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The next theorem gives a divergence result equal to that in Theorem 2. However, the preconditions are different and no statement about the divergence speed is made in Theorem 3. In Theorem 2 we need the additional condition (15), which is of course restrictive. It is interesting to analyze the situation if the the lower and upper bound from Lemma 1 can be coupled.

Theorem 3. Let {[t.sub.k]}.sub.k[member of]Z] be defined as in Lemma 1 and [phi] as in equation (8). Furthermore, let [phi] [member of] [B.sup.[infinity].sub.[phi]]. If there is a constant [C.sub.10] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

for all N [member of] N, then there exists a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

Remark 3. Note that the requirement (17) is, for example, fulfilled for [phi](t) = sin([pi]t).

Proof. From equation (17) and (9) we know that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assume that there is no signal f [member of] [PW.sup.1.sub.[pi]] which satisfies equation (18). Then by the Banach-Steinhaus theorem there is a constant C12 such that

[parallel][S.sub.N][parallel] [less than or equal to] [C.sub.12] (19)

for all N [member of] N. Therefore,

[N.summation over (k=-N)][absolute value of [phi](t)]/[absolute value of [phi]'([t.sub.k])] [absolute value of t - [t.sub.k]] [less than or equal to] [C.sub.13]

holds for all N [member of] N, t [member of] R and

[[infinity].summation over (k=-[infinity])][absolute value of [phi](t)]/[absolute value of [phi]'([t.sub.k])] [absolute value of t - [t.sub.k]] [less than or equal to] [C.sub.13] (20)

all t [member of] R, because [C.sub.13] is independent of N. Let [epsilon] > 0 be arbitrary and consider the function [phi]((1 - [epsilon])t), t [member of] R, which is obviously in [B.sup.[infinity].sub.(1-[epsilon])[pi]]. Moreover,

[g.sub.[epsilon]](t) = [phi]((1 - [epsilon])t) sin([pi][epsilon]t)/[phi][epsilon]t, t [member of] R,

is a signal in [PW.sup.2.sub.[pi]] and consequently

[g.sub.[epsilon]](t) = [[infinity].summation over (k=-[infinity])] [g.sub.[epsilon]](t)(t.sub.k][[phi].sub.k](t)

for all t [member of] R. Let t [member of] R, t [not equal to] [t.sub.k], k [member of] Z arbitrary but fixed and [delta] > 0 arbitrary. Due to equation (20) there is a [k.sub.0] = [k.sub.0]([delta]) such that

[summation over ([absolute value of k)[greater than or equal to][k.sub.0])] [absolute value of [[phi].sub.k](t)][less than or equal to] [delta].

This leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Inequality (21) is valid for all [epsilon] > 0, but the right-hand side of (21) is independent of [epsilon]. There is a [[epsilon].sub.0] = [[epsilon].sub.0]([delta]) such that

[absolute value of [phi](t) - [g.sub.[epsilon]](t)] < [delta]

and

[absolute value of [phi]([t.sub.k]) - [g.sub.[epsilon]]([t.sub.k])] = [g.sub.[epsilon]]([t.sub.k]) < [delta], [absolute value of k] [less than or equal to] [k.sub.0]

holds for all [epsilon] [less than or equal to] [[epsilon].sub.0]. Consequently, for all 0 < [epsilon] [less than or equal to] [[epsilon].sub.0]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [delta] > 0 is arbitrary, it follows that [phi](t) = 0. But we assumed t [not equal to] [t.sub.k], k [member of] Z, which leads to a contradiction. Thus, (19) cannot hold and (18) must be true.

The Kadec 1/4-Theorem [28, p. 42] answers the question when the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] forms a Riesz basis for [L.sup.2][-[pi], [pi]].

Theorem 4 (Kadec's 1/4-Threorem). If {[t.sub.k]}.sub.k[member of][Z]. is a sequence of real numbers for which

[absolute value of [t.sub.k] - k] [less than or equal to] [bar.[delta]] < 1/4, k [member of Z,

then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] forms a Riesz basis for [L.sup.2][-[pi], [pi]].

It is remarkable that for [bar].[delta]] < 1/4 the discussed reconstruction process is uniformly convergent on R for signals in [PW.sup.2.sub.[pi]], and for signals in [PW.sup.1.sub.[pi]] not even uniformly bounded if the functions [phi] fulfill certain conditions. Furthermore, the considered set of sampling points is minimal in the sense that the removal of one point eliminates the uniqueness in the interpolation problem. We will see in Section 4 that, as soon as the assumption of minimality is dropped, a globally, uniformly convergent reconstruction is possible even for [PW.sup.1.sub.[pi]].

In this section we discussed sampling series of the shape

[N.summation over (k=-N)] f([t.sub.k])[[phi].sub.k](t), t [member of] R, (22)

where the points [{[t.sub.k]}.sub.k[member of]Z] formed a complete interpolating sequence for the space [PW.sup.2.sub.[pi]]. It was shown that for signals f [member of] [PW.sup.1.sub.[pi]] the sampling series are neither globally uniformly convergent nor uniformly bounded on R, in general. In order to show this, it was important to require minimality and to pose restrictions on the bandwidth of the reconstruction process. In the next section it is shown that a global uniform convergence is possible if minimality and minimal bandwidth are waived. However, it is not clear whether oversampling and a larger bandwidth are really necessary in order to achieve a locally uniformly convergent and globally uniformly bounded reconstruction of all signals f [member of] [PW.sup.1.sub.[pi]]. We suppose that this is indeed necessary, i.e., our conjecture is as follows.

Conjecture 1. For every complete interpolating sequence [{[t.sub.k]}.sub.k[member of]Z] for [PW.sup.2.sub.[pi]] there exists a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [phi] defined as in equation (8).

Even if (22) is not uniformly convergent on whole R, it might be uniformly convergent on compact subsets of R. If this is true does not only depend on f but also on the properties of the functions [[phi].sub.k], k [member of] Z. For [[phi].sub.k](t) = sin([pi] (t - k))/([pi](t-k)) and [t.sub.k] = k, k [member of] Z we know from Theorem 1 that (22) is uniformly convergent on compact subsets of R for all f [member of] [PW.sup.1.sub.[pi]]. However, we think that there are Riesz bases where (22) does not converge uniformly on compact subsets of R to f.

Conjecture 2. For all T > 0 there exists a complete interpolating sequence [{[t.sub.k]}.sub.k[member of]Z] for [PW.sup.2.sub.[pi]] and a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [phi] is defined as in equation (8).

Conjecture 2 seems to be difficult to prove. However, we have been able to obtain a weaker result. In [3] we proved that if [{[t.sub.k]}.sub.k[member of]Z] is a complete interpolating sequence for [PW.sup.2.sub.[pi]], then for all t [member of] R there exists a stable linear time-invariant system [T.sub.1] and a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

This is good evidence that the Conjecture 2 is true.

4 Signal Reconstruction with Oversampling

In this section the stability of reconstruction processes with oversampling is analyzed. Oversampling creates a degree of freedom in the choice of the reconstruction kernel [13]. If the kernel is chosen appropriately, then the reconstruction process is uniformly convergent on the whole real axis [5].

As we will see by Theorem 5, even the Shannon sampling series is uniformly convergent on whole of R when oversampling is applied. In order to prove Theorem 5 we need the results from Lemma 2 and 3. However, for better readability we defer the lemmas and begin with Theorem 5.

Theorem 5. Let a > 1 be fixed. Then we have for all f [member of] [PW.sup.1.sub.[pi]] that

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

Proof. Let f [member of] [PW.sup.1.sub.[pi]] be arbitrary but fixed and [member of] > 0. Then there is a [f.sub.1] [member of] [PW.sup.2.sub.[pi]]. such that [f.sub.1](k) is different from zero only for finitely many k [member of] Z and it holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Obviously, [f.sub.1] [member of] [PW.sup.2.sub.[pi]] [subset] [PW.sup.2.sub.[alpha][pi]]. Therefore, there exists a [N.sub.0] = [N.sub.0]([epsilon]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all N [greater than or equal to] [N.sub.0]. Moreover, for all N [greater than or equal to] [N.sub.0],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

where Lemma 2, applied to [f.sub.1] - f, has been used for the last inequality. Since (24) holds for all [member of] > 0, the proof is complete.

A crucial part in the proof was Lemma 2, which can be used to analyze the influence of the oversampling on the peak value of the partial sum of the Shannon sampling series. It is interesting to note, that the 1/cos([pi](2a)) term from equation (1) reappears as integral part of Lemma 2.

Lemma 2. Let a > 1 be fixed. Then we have for all f [member of] [PW.sup.1.sub.[pi]] and all N [member of] N

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

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where we introduced the abbreviation

[g.sub.N](t, [omega], a) := [N.summation over (k=-N)] [e.sup.i[omega]k/a] sin(a[pi](t - k/a))/a[pi](t - k/a)

and used Lemma 3 for the last inequality.

Lemma 3. For all a > 1, t, [omega] [member of] R, [absolute value of [omega]] [less than or equal to] [pi], N [member of] N it holds that

[absolute value of [g.sub.N](t,[omega],a)] [less than or equal to] 2 (1 + 2/cos([pi]/2a)).

Proof. Only the case t > 0 needs to be analyzed. For t = 0 we have [g.sub.N](0,[omega],a) = 1 and the case t < 0 is treated analogously to the case t > 0. If t [member of] A = {k/a: k [member of] N}, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and therefore [absolute value of [g.sub.N](t,[omega],a)] [less than or equal to] 1. Hence, t > 0, t [not member of] A can be assumed. Let N [member of] N be arbitrary but fixed. Obviously,

[g.sub.N](t,[omega],a) = sin(a[pi]t)/a[pi] [N.summation over (k=-N)] [e.sup.ik([omega]/a+[pi])] [1]/[t - k/a] = sin(a[pi]t)/a[pi] [N.summation over (k=-N)] [c.sub.k] [d.sub.k], (26)

where [c.sub.k] [e.sup.ik([omega]/a+[pi])] and [d.sub.k] = 1/(t- k/a). Let [C.sub.k] = [[summation].sup.k.sub.l=1N][c.sub.l], [absolute value of k] [less than or equal to] N. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We begin with the case t > (N + 1)/a. Summation by parts gives

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

The right-hand side of (27) can be further simplified by evaluating the telescoping series

[N-1.summation over (k=-N)] (1/t - k+1/a - 1/t - k/a) = (1/t - N/a - 1/t + N/a [less than or equal to] a (28)

for t > (N + 1)/a. Combining equations (26), (27), and (28) leads to

[absolute value of [g.sub.N](t,[omega],a)] [less than or equal to] 2/[pi] cos([pi]/2a).

Next, the case t < (N + 1)/a is treated. Let [N.sub.t] be the largest natural number such that [N.sub.t]/a < t. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

The first term on the right-hand side of equation (29) can be upper bounded by

[absolute value of [N.sub.t]-1.summation over (k=-N)] [c.sub.k][d.sub.k]] [less than or equal to] 2a/cos([pi]/2a)

exactly in the same way as before and the second term by

[absolute value of N.summation over (k=[N.sub.t]+2)] [c.sub.k][d.sub.k]] [less than or equal to] 2a/cos([pi]/2a).

Theorem 5 shows that if oversampling is used, we can have global uniform convergence for all f [member of] [PW.sup.1.sub.[pi]]. An interesting question concerns the rate of convergence of the sampling series (1): Given some a > 1, can we find two constants [gamma] = [gamma](f, a) > 0 and C = C(f, a) < [infinity] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every f [member of] [PW.sup.1.sub.[pi]]? Is it even possible to find a [gamma], independently of f? The answer to both questions is given by the next theorem.

Theorem 6. Let a > 1 be fixed. For each arbitrary sequence {[[epsilon].sub.N]} of positive numbers that converges to zero, there exists a f [member of] [PW.sup.1.sub.[pi]] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. In order to accomplish the proof, we introduce the operator

([R.sub.N] f)(t) := f(t) - [N.summation over (k=-N) f (k/a) sin(a[pi](t - k/a))/a[pi](t - k/a).

Obviously we have

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

Since (30) is valid for all t [member of] R, we can choose t = (N + 1)/a. Then we have

[N.summation over (k=-N)] [e.sup.i[omega]k/a] sin(a[pi](t - k/a))/a[pi](t - k/a) = 0

and consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, the operator [[??].sub.N] [member of] N, defined by

([[??].sub.N]f)(t) := 1/[[epsilon].sub.N]([R.sub.N]f)(t),

is linear and bounded, and we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, by the Banach-Steinhaus theorem there is a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that lim [sup.sub.N[right arrow][infinity]] [[parallel][[??].sub.N][f.sub.1][parallel].sub.[infinity]] = [infinity], which completes the proof.

Theorem 6 shows that the convergence speed of the Shannon sampling series with oversampling for the space [PW.sup.1.sub.[pi]] can be arbitrarily slow and that no convergence rates can be given.

5 Non-Symmetric Sampling Series

Many attempts have been made to proof convergence results that are true for the symmetric sampling series of the non-symmetric case. However, it turned out that the non-symmetric sampling series exhibits a significantly different convergence behavior compared to the symmetric sampling series for certain problems. This is particularly true for the Shannon sampling series and the space [PW.sup.1.sub.[pi]]: It has been shown [1, 4] that there is a signal f [member of] [PW.sup.1.sub.[pi]] such that for all t [member of] R/Z

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which means that the non-symmetric Shannon sampling series does not converge to f for all t [member of] R\Z. Consequently, a theorem stating the uniform convergence of the non-symmetric Shannon sampling series on compact subsets of R, similarly to Theorem 1, which states exactly this for the symmetric Shannon sampling series, of course cannot exist.

The preceding considerations have been made for the case where the samples are taken at Nyquist rate, i.e., where no oversampling is applied. The convergence behavior of the non-symmetric Shannon sampling series changes completely if we use oversampling. A closer look at the proof of Lemma 3 shows that the inequality convergence behavior of the non-symmetric Shannon sampling series changes completely if we use oversampling. A closer look at the proof of Lemma 3 shows that the inequality.

[N.summation over (k=-M)] [e.sup.i[omega]k/a] sin([alpha][pi](t - [k/[alpha]))/[alpha][pi](t - k/[alpha]]) [less than or equal to] 2 (1 + [2/cos([pi]/2[alpha]))

holds for all M, N [member of] N, t [member of] R and [absolute value of [omega]] [less than or equal to] [pi]. Therefore, it is possible to get an analogous result to Lemma 3 for the non-symmetric sampling series. Consequently, a theorem similar to Theorem 5 can be derived for the non-symmetric sampling series:

Theorem 7. Let a > 1 be fixed. Then we have for all f [member of] [PW.sup.1.sub[pi]]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

Proof. Analogously to Theorem 5.

We have seen that for f [member of] [PW.sup.1.sub.[pi]] there are differences in the convergence behavior between the symmetric Shannon sampling series and the non-symmetric Shanon sampling series when no oversampling is applied. However, with oversampling both have the same good convergence behavior, i.e., both are uniformly convergent on whole of R.

6 Convergence Analysis

We return to the discussion begun in Section 3, namely the question of finding a sufficient condition for uniform convergence of the reconstruction process. The results from Section 4 can be used to analyze the convergence behavior of the Shannon sampling series for [PW.sup.1.sub.[pi]] signals without oversampling. It has been proven that, for some f [member of] [PW.sup.1.sub.[pi]], the Shannon sampling series

[[infinity].summation over (k=-[infinity])] f(k) sin([pi](t - k))/[pi](t - k)]

does not converge uniformly on whole of R to f. However, if [??]([omega) fulfills certain integrability conditions in the vicinity of [omega] = [pi], even the non-symmetric Shannon sampling series converges uniformly on whole of R.

Theorem 8. If f [member of] [PW.sup.1.sub.[pi]] has the property that there exists a [delta] > 0 and a p > 1 such that

[[infinity].sub.[pi]-[delta][less than or equal to][absolute value of [omega]] [less than or equal to][pi]] [[absolute value of [??]([omega])].sup.p] d[omega] < [infinity],

then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Two auxiliary bandlimited signals [f.sub.1] and [f.sub.2] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are needed for the proof. Obviously, [f.sub.1] [member of] [PW.sup.1.sub.[pi]-[delta]] and f(t) = [f.sub.1](t) + [f.sub.2](t), t [member of] R. Furthermore, it holds by assumption that

1/2[pi] [[infinity].sup.[pi].sub.-[pi]] [[absolute value of [[??].sub.2]([omega])].sup.p] d[omega] < [infinity].

Therefore, [f.sub.2] [member of] [PW.sup.p.sub.[pi]]. Let [epsilon] > 0 be arbitrarily chosen. Then there exists a [N.sub.0] = [N.sub.0]([epsilon]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all N [less than or equal to] [N.sub.0]. Since [f.sub.1] [member of] [PW.sup.1.sub.[pi]-[delta]], there exists a [N.sub.1] = [N.sub.1]([epsilon]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all N [greater than or equal to] [N.sub.1]. Consequently, for all N [greater than or equal to] max([N.sub.0], [N.sub.1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

Since inequality (32) is valid for all t [member of] R, the proof is completed.

Theorem 8 shows that the known divergence of the Shannon sampling series for signals f [member of] [PW.sup.1.sub.-1[pi]] is only a consequence of the behavior of [??]([omega]) in the vicinity of [omega] = [pi]. If certain integrability conditions are fulfilled in this region, then the divergence does not appear.

7 Oversampling and Kernels

In this section we want to examine whether oversampling is really a universal remedy for circumventing convergence problems. We begin with the following observation: The finite sampling series

1/a[pi] [N.summation over (k=-N) f(k/a) sin(a[pi](t - k/a]))/t - k/a (33)

with oversampling factor a > 1 is not bandlimited to [pi], but rather to a[pi]. Can we use functions in the approximation formula that are bandlimited to [pi] itself?

For f [member of] [PW.sup.2.sub.[pi]] this is obviously possible. By expanding [??] into a Fourier series in the interval [-a[pi], a[pi]], a > 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, using the definition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of the characteristic function [[chi].sub.[pi]], it holds that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where we used the Cauchy-Schwarz inequality. Therefore,

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

As a consequence, for f [member of] [PW.sup.2.sub.[pi]] it is possible to use

1/a[pi] [N.summation over (k=-N)] f(k/a) sin([pi](t - k/a))/t - k/a (35)

for signal reconstruction. This has the advantage that f [member of [PW.sup.2.sub.[pi]] can be approximated according to equation (35) by a signal, which is bandlimited to [pi]. The sampling series (35) can be obtained by low-pass filtering the signal generated by the sampling series in equation (33) with bandlimit [pi].

Since for all signals f [member of] [PW.sup.1.sub.[pi]], the series in (33) converges uniformly on whole of R to the signal f, it is reasonable to ask whether a low-pass filtering of (33) preserves the uniform convergence, even for f [member of] [PW.sup.1.sub.[pi]]. Then, this would be the projection on the desired frequency interval. However, the following theorem gives a negative answer.

Theorem 9. There is a [f.sub.1] [member of] [PW.sup.1.sub.[pi]] such that

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

or equivalently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. For t [member of] R, a > 1, N [member of] N fixed, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next,

[q.sub.N](t, [omega], a) := [N.summation over (k=-N)] [e.sup.i[omega]k/a] sin([pi](t - k/a))/a[pi](t - k/a)

is analyzed. For [omega] = [pi] and [t.sub.N] = (N + 1/2)/a we get

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

The second sum in equation (37) was evaluated in the same way as the sum in (26). Consequently, for [t.sub.N] = (N + 1/2)/a we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, by the Banach-Steinhaus theorem there is a signal [f.sub.1] [member of] [PW.sup.1.sub.[pi]] that fulfills (36).

The result of Theorem 9 is interesting: it shows that a low-pass filtering destroys the uniform convergence. Not even the uniform boundedness is preserved. This means that the sole redundancy in the set [{f(k/a)}.sub.k[member of]Z], a > 1, is not sufficient for uniform convergence. It is necessary to use a proper kernel. Consequently, a simultaneous projection into the space [PW.sup.1.sub.[pi]] during reconstruction is not possible without the reconstruction process to become divergent.

It is easy to see that for f [member of] [PW.sup.1.sub.[pi]] no reconstruction process of the shape

[N.summation over (k=-N)] f (k/a) [phi] (t - k/a) (38)

and with the property that (38) is bandlimited to [pi] can be uniformly bounded. If (38) is bandlimited to [pi], then the Fourier transform

1/a [N.summation over (k=-N)] f(k/a) [e.sup.i[omega]k/a](a[??]([omega]))

is supported in [[pi], [pi]]. Since [[summation].sup.N.sub.k=-N] f(k/a)[e.sup.iwk/a] has only isolated zeros, [??]([omega]) = 0 for [absolute value of [omega]] > [pi] must hold. Moreover, the assumed convergence of (38) implies that a[??]([omega]) = 1 for [absolute value of [omega]] < [pi]. Hence, [phi](t) = (sin([pi]t))/(a[pi]t). But Theorem 9 has shown that the reconstruction process for this kernel is neither uniformly convergent nor uniformly bounded on whole of R.

We have seen that it is impossible to have a locally uniformly convergent and globally, uniformly bounded reconstruction for all f [member of] [PW.sup.1.sub.[pi]] on the basis of the samples [{f(k/a)}.sub.k[member of]Z] if the reconstruction process is of the shape (38) and [phi] is bandlimited to [pi]. It is possible to approach the bandlimit [pi] arbitrarily closely, but to have [phi] exactly bandlimited with [pi] is impossible if uniform boundedness is desired.

The perception that oversampling is useful for almost everything is prevalent. Indeed, for all signals f [member of] [PW.sup.1.sub.[pi]] a uniformly convergent reconstruction is possible if the signal is known on an oversampling set {f(k/a)}, a > 1. Consider [omega] with a[??]([omega]) = 1 for [absolute value of [omega]] [less than or equal to] [pi], [??]([omega]) = 0 for [absolute value of [omega]] [greater than or equal to] a[pi] and [phi] [member of] [B.sup.1.sub.a[pi]]. Then we have for all f [member of] [PW.sup.1.sub.[pi]]

f(t) = [[infinity].summation over (k=-[infinity]) f(k/a) [phi] (t - k/a),

and since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the sampling series is absolutely convergent.

ACKNOWLEDGEMENT

The authors would like to thank J. Hagenauer for discussions on earlier versions of this paper.

This work was partly supported by the German Research Foundation (DFG) under grant BO 1734/9-1.

The material in this paper was presented in part at the 7th International ITG Conference on Source and Channel Coding (SCC'08), Ulm, Germany, January 2008.

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Holger Boche

Heinrich-Hertz-Chair for Mobile Communications

Technische Universitat Berlin

Einsteinufer 25

D-10578 Berlin, Germany

holger.boche@mk.tu-berlin.de

Ullrich J. Monich

Heinrich-Hertz-Chair for Mobile Communications

Technische Universitat Berlin

Einsteinufer 25

D-10578 Berlin, Germany

ullrich.moenich@mk.tu-berlin.de

(1) This question was asked by one of the reviewers.

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Author: | Boche, Holger; Monich, Ullrich J. |
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Publication: | Sampling Theory in Signal and Image Processing |

Article Type: | Report |

Date: | Jan 1, 2009 |

Words: | 9394 |

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