# Groups and the sampling theorem.

The symmetries that underlie Shannon's sampling theorem and its more general multi-band version are used as a basis for an exposition of sampling theory in a locally compact abelian group setting. Kluvanek's theorem, which is an extension of Shannon's sampling theorem to this setting, together with a converse, are proved and a connection with some ideas in probability is sketched. Some applications are obtained by specialising the abstract groups to particular cases, such as the real line or the circle group.

Key words and phrases: Whittaker-Kotel'nikov-Shannon theorem, Plancherel's formula, locally compact abelian groups, discrete subgroups, tranvsersals

2000 AMS Mathematics Subject Classification--94A05, 42A99

1 Introduction

An analogue signal f can be thought of as outputting a value at each instant in the time domain. Time is taken to be the real line, denoted by R, and the output of f will typically (but not necessarily) be another real number. In mathematical terminology, f is a continuous real-valued function of real numbers, written concisely in mathematical notation as f: R [right arrow] R. While in engineering it might not make much sense, there is nothing to lose and much to be gained by allowing f to take complex values, i.e., to consider f: R [right arrow] C, where C denotes the complex numbers. The frequency domain is also the real line R and the Fourier transform f^ of f can be regarded as a function f^: R [right arrow] C. The title of this paper comes from ideas of symmetry which underlie the Shannon sampling formula. As is well known, the sampling formula provides a reconstruction of band-limited signals (i. e., signals with frequencies [omega] at most W) from regularly spaced samples (accounts of this result, its applications and its manifold extensions can be found in [20, 21, 23]). For completeness the result, often known as the Whittaker-Kotel'nikov-Shannon (WKS) theorem, is now stated formally.

Theorem. Let f: R [right arrow] C be a continuous and square integrable function (or analogue signal) with Fourier transform

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which vanishes outside [-W, W]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where sinc x = (sinx)/x.

An observation which might seem trite is that the translates [-W, W] + 2W k for integers k of the spectrum [-W, W] of the signal f intersect only at endpoints (see Figure 1).

[FIGURE 1 OMITTED]

However, the spectral repetition in the frequency domain by translates allows the real line R to be tesselated by the intervals, as in Figure 1. Evidently these copies of the Fourier transform form a 2W-periodic function on R and this periodic function has itself a Fourier series. This is the basis of the proof of the Sampling Theorem, shown diagrammatically in Figure 2.

[FIGURE 2 OMITTED]

The Fourier transform (spectrum) f^ of the function f in (a) is indicated in (b). The spectral repetition in (c) is a periodic function with discrete Fourier transform shown in (d). Starting with the samples in (d), one takes the inverse Fourier transform, as shown in (c), and then removes the repetitions by means of the characteristic or indicator function X[-W,W] for the interval [-W,W] (see (b)), i.e.,

X[-w,w]([omega]) = 1, if -W [less than or equal to] [omega] [less than or equal to] W, 0, otherwise. (2)

This function acts as a filter; on taking the inverse Fourier transform of the filtered bandpass signal in (b), one recovers the function or analogue signal f in (a), given by (1).

A more general 'multi-band' theory [5, 10, 26, 27], in which the interval [-W,W] is replaced by a set A satisfying the condition that distinct translates of the spectrum are 'almost disjoint' (or, in precise mathematical language, the translates intersect in a set of Lebesgue measure 0)--see Figure 3. Although of independent interest, this theory has applications in signal processing (see for example ).

[FIGURE 3 OMITTED]

Instead of being band-limited, the functions (signals) are square-integrable (i. e., [integral] R [|f (x)|.sup.2] dx exists), continuous and with Fourier transforms which vanish outside a set A satisfying | A [intersection] (A + k/s)| = 0, s > 0 for all integers k [not equal to] 0 (|A| is the Lebesgue measure or 'length' of A). Such functions have the representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [X.sup.V.sub.A] is the inverse Fourier transform of the characteristic function of the set A. This formula is of use in dealing with band-pass signals  and its validity implies essentially that translates by 1/s of the spectrum are disjoint . The parameter s is the distance between samples (s = 1/(2W) in (1)) and is at most 1/|A|. The sampling rate is 1/s and is at least the Nyquist rate of twice the maximum frequency but can be considerably less for suitable spectra. In Figure 1 and in (1), |A| = 2W, whence the Nyquist rate of twice the maximum frequency is 2W. In Figure 3, where W = 1/2, s = |A| = 1, the Nyquist rate is 3 (twice the maximum frequency of 1.5) but the spectrum has been chosen to allow the most efficient sampling rate of 1. In general, however, there will be gaps, and the sampling rate will be greater than optimal and higher than the length of the spectrum.

The tessellations or translates which play such a key role reveal the importance of the group structure in the Sampling Theorem. In this paper, the theorem will be looked at in an abstract group setting, where the groups replacing the real numbers are abelian and locally compact, and so enjoy a 'measure' or generalised area. This is enough to carry the mathematics needed for a theory closely (but not completely) analogous to the familiar classical sampling theory, including the multi-band theory and irregular sampling theory. (The abstract version of this important area will not be discussed here but see  and the references therein.)

The symbol [T.sup.1] stands for the circle group {z [member of] C: |z| = 1} or one-dimensional torus; the symbol [S.sup.1], representing the one-dimensional sphere, is also used but not here to avoid possible confusion with sampling. These very different groups R, [T.sup.1], Z are all abelian and locally compact ([T.sup.1] is compact), and so are united by working with locally compact abelian groups.

2 The abstract setting

Abstract harmonic analysis is the study of Fourier analysis on abstract groups. When the groups are not abelian, the topic is very difficult, involving exotic mathematics such as representation theory and, apart from finite groups (a brief but accessible introduction is in [25, [section] 104], more details can be found in for example [22, 32]), rather demanding. For abstract harmonic analysis on some special types of infinite groups, the text  is an attractive introduction but requires a fair amount of background. As far as I am aware there has been little progress in developing a sampling theory in the non-abelian group setting.

However we shall be concerned with locally compact abelian groups, where the theory has close analogues with the classical theory and is particularly well suited to sampling theory. For example, the exponential [e.sup.2[pi]itw] is replaced by a so-called 'character' (x, [gamma]) which (like the exponential) is a point with modulus 1 on the unit circle and with the same multiplicative properties, i.e.,

(x + x', [gamma]) = (x, [gamma])(x', [gamma]) and (x, [gamma] + [gamma]') = (x, [gamma])(x, [gamma]'). (4)

This behaviour is one of the cornerstones for classical theory, so that the abstract theory has much in common with the classical theory. An abstract version of the inversion theorem holds (although with some additional restrictions); there is an analogue of Plancherel's theorem and the analogues of the related Parseval and convolution theorems can be deduced in a manner similar to the classical case.

The locally compact abelian group framework has two advantages, one general and the other specific. First, it unites a number of disparate results into one general framework with a concise and elegant notation and, secondly, it distinguishes explicitly between the time and frequency domains of a signal, revealing some interesting reciprocity relations. Regarding the former, a number of different versions of the theorem can be obtained from the one abstract result simply by choosing the appropriate underlying group. Examples are given in , where versions of the Sampling Theorem are obtained for the real line R, the circle [T.sup.1], the torus (or the tube of a car-tire) [T.sup.2], the integers Z, the dyadic group and Euclidean space [R.sup.n].

Some essential preliminaries and notation are now given. More details can be found in Rudin's monograph , in the comprehensive volumes of Hewitt and Ross [17, 18], or in .

2.1 Locally compact abelian groups

An abelian group G has an operation of addition denoted by + that is commutative, i.e., for each a,b in G, a + b = b + a, lies in G and is associative ((a+b) + c = a + (b+c), i.e., it does not matter in which order pairs are added), a zero element (or additive identity 0 (a + 0 = a)) and an additive inverse: the inverse of a [member of] G is -a and a + (-a) = 0 (a + (-b) is written a - b). If G is a topological group as well, then it has a notion of 'closeness' or how near points are. The topology on the group is related to the idea of 'distance' between two points, but is more general and disregards continuous reversible deformations such as stretching (but not tearing). From a topological point of view, a coffee cup and a doughnut are identical.

A compact topological group, such as the circle group [T.sup.1] = {z [member of] C: |z| = 1}, is roughly speaking finite dimensional and has some nice properties (often enjoyed by finite sets--in fact, compactness can be thought of as a generalisation of finiteness). Finite groups such as {0, 1, ..., k - 1}, the residues of integers modulo k, are compact. A locally compact group is compact on a small scale. A compact group is automatically locally compact. The real line R, which is the model for this theory, has each of these properties; it includes the closed unit interval [0, 1] which is compact. Other examples are Euclidean space, C, the circle [T.sup.1], and the n-dimensional torus [T.sup.n]. The integers Z form a discrete group in which the elements are separated and is a locally compact group.

A subgroup H of G is a subset of G which is a group in its own right; thus, H contains the zero element 0, and is closed under addition (for any h, h' [member of] H, the sum h + h' is defined and lies in H) and -h [member of] H. Subgroups have nice properties and a closed subgroup of a locally compact abelian group is itself a locally compact abelian group, with Haar measure [m.sub.H].

The translate H + a = {h + a: h [member of] H} of the subgroup H by an element a [member of] G is called a coset and is often written H + a = [a]. The coset H + a is either H itself or disjoint from H. In fact, if a [??] H, then H and the translate of H by a are disjoint: H [intersection] (H + a) = [??], while if a [member of] H, H + a = H. Moreover, the cosets have a group structure of their own: two cosets H + a, H + a' are added by adding the cosets, as follows,

(H + a) + (H + a') = H + (a + a').

As well, H + (H + a) = H + a, so H(= H + 0) itself behaves as an additive identity or zero element for the set of cosets. The group of cosets {H+a: a [member of G} is called the quotient group and is written G/H. The group G/H 'absorbs' the structure of H and is simpler than G. The commutativity of addition is crucial and the situation is more complicated for non-abelian groups. More details can be found in books on group theory, e.g.,  (despite being rather old, this is an excellent introduction, though 'Haar' is spelt 'Harr').

Given a coset H + a, the element a can be chosen as a representative of the coset. This choice is not unique and any other element a' = a + h for some h [member of] H would also serve. A set K formed by a collection of such representatives is called a transversal, and by definition K consists of one and just one point from each distinct coset H + a = [a] (the transversal is also called a complete set of coset representatives but we shall stick to the shorter name). Thus, K [intersection] (H + a) consists of a single point in K and translates of H by non-zero elements in H are disjoint. For example, [-W, W) and [0,2W) are both transversals for the quotient group R/(2WZ), where 2WZ is the set of all 2Wk, k [member of] Z (note that W and -W cannot both be in the transversal). Later it will be convenient to consider [-W, W], the transversal [-W, W) with the additional point W added but this will make no difference to the calculations.

When H is a topologically closed subset of G (a technicality that is satisfied in all that follows), G/H is also a locally compact abelian group, with Haar measure [m.sub.G/H], and plays an important role in the abstract theory. The measures [m.sub.G], [m.sub.H], and [m.sub.G/H] are related, as we shall see in the next subsection.

It is a remarkable fact that the property of closeness combined with good local behaviour tell us that the locally compact abelian group G has two important features. First, it has an essentially unique Haar measure and, secondly, it is associated with a dual group.

2.2 Haar measure and integral

Haar measure is a very general extension of a 'length' or 'area', which it turns out is in many respects like Lebesgue measure (a mathematical name for ordinary length, area or volume in Euclidean space). It is unique up to a multiplicative constant which is usually fixed by a convenient normalisation or choice of a unit. The Haar measure of a set E in G will be denoted by [m.sub.G](E) and it has the pleasing property that it is invariant under translation by elements in the group, i.e., [m.sub.G](E + a) = [m.sub.G](E), just as length and area are unchanged by translation. The Haar measure for [R.sup.n] can be taken to be n-dimensional Lebesgue measure, that of the discrete group Z, counting measure and that of the compact circle group [T.sup.1] = {z [member of] C: |z| = 1}, arc length. An important consequence is that an integral over G can be defined, with properties very like the usual or Lebesgue integral. In particular, the integral is translation invariant, i.e., for a given [a.sub.0] [member of] G,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For suitable f (technically for integrable f or f [member of] [L.sup.1](G), where the 'L' stands for Lebesgue who developed the modern theory of integration), the Haar integral is denoted by [[integral].sub.G] f(a)[dm.sub.G](a) or more concisely by [[integral].sub.G] f. Again as in the classical case, functions f fall into sets [L.sup.p](G), where 1 [less than or equal to] p < [infinity], according to the existence of the integral [integral] [|f|.sup.p]. We shall be concerned only with p = 1, 2. Sometimes we write [[integral].sub.G] [|f|.sup.2] = [||f||.sup.2.sub.G] ([||f\\.sub.G] represents the 'length' of f in [L.sup.2](G)).

Intervals, such as [0, 1] or [-[pi], [pi]], lie within the real line R. This enables the relationship between Fourier series (periodic functions) and transforms in sampling theory to be developed naturally. Locally compact abelian groups do not necessarily have this sort of structure, but the Haar measures on H or G/H will be normalised so that the fundamental coset decomposition formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

holds for suitable functions f (see [18, [section] 28.54 (iii)] and [30, [secton] 2.7.3]).

Note that the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is constant on cosets, since for each h' [member of] H,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This means that we can define a function [??]: G/H [right arrow] C on cosets H + a given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In fact, every function in [L.sup.1] (G/H) is of this form [18, [section] 28.54].

When G is discrete and countable, each point a has a non-zero measure or 'mass', written [m.sub.G]({a}) = [m.sub.G]({0}) by translation invariance. The uniqueness of Haar measure implies that the integral over G of a function f (in [L.sup.1](G)) reduces to a constant multiple of a sum, so that for each f [member of] [L.sup.1](G),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus, when H is discrete and countable, the sum

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and is constant on cosets H + a. All sets discussed will be assumed to be Haar measurable, a mathematical technicality that is unimportant in practice.

2.3 The dual group

The dual group of G is denoted by G^ or by [GAMMA] (used from henceforth) and consists of the group of continuous characters [gamma]: G [right arrow] [T.sup.1] which satisfy

[gamma] (a + b) = [gamma](a)[gamma](b), a,b [member of] G.

The characters are abstractions of the exponential [e.sup.2[pi]itw]. The dual group [GAMMA] is also locally compact and abelian, with its Haar measure denoted by [m.sub.[GAMMA]], and written additively by defining the sum of two elements [gamma], [gamma]' as ([gamma] + [gamma]')(a) = [gamma](a)[gamma]'(a). In view of the duality between G and [GAMMA], the unimodular complex number [gamma](a) [member of] [T.sup.1] is usually written (a, [gamma]). Note that (0, [gamma]) = 1 = (a, 0) and that (a, [gamma]) (a, [gamma]) = 1. The dual G^^ of the dual group G^ of the locally compact abelian group G is isomorphic to G; the dual of a compact abelian group is discrete and vice-versa. The picture is very different in the non-abelian case as the characters have to be replaced by representations, which are homomorphisms to certain (non-abelian) automorphism groups, and the dual group does not necessarily exist (for more details see [17, 18, 33]).
```Table 2 Dual groups

Group Dual

R R
[T.sup.1] Z
Z [T.sup.1]
[T.sup.1] x R Z x R
[Z.sub.k] [Z.sub.k]
```

In this approach, the dual group [GAMMA] corresponds to the frequency domain of a signal and so is very much in the foreground. Continuous functions in the Hilbert space

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

correspond to analogue signals with finite energy.

2.4 The Fourier transform

The Fourier transform is modelled on the classical definition: the Fourier transform of the function f: G [right arrow] C in [L.sup.1](G) will be denoted by f^: [GAMMA] [right arrow] C and is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The inverse Fourier transform [[psi].sup.V]: G [right arrow] C of each [psi] in [L.sup.1] ([GAMMA]) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and is continuous.

We need to consider the Fourier transform and the inverse Fourier transform (which exist for integrable ([L.sup.1]) functions) of square integrable ([L.sup.2]) functions of G or [GAMMA]. The relationship between these types of functions is rather complicated. Square integrable functions f: G [right arrow] C that are also integrable are dense in [L.sup.2](G), i.e., any [L.sup.2] function is 'close' to a function which is [L.sup.1] and [L.sup.2]. This allows the Fourier transform of an [L.sup.2] function to be defined as a limit of approximating functions which are integrable and square integrable. We shall not go into these technicalities (covered in detail in [17, 18, 30]), as the engineering origins of the Sampling Theorem make them largely irrelevant.

The Haar measure [m.sub.[GAMMA]] on the dual group [GAMMA] of G is normalised so that for suitable f the Inversion Theorem holds [30, [section] 1.5], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The abstract version of Plancherel's theorem considers the Fourier transform [LAMBDA] as an operator from [L.sup.2](G) to [L.sup.2]([[GAMMA]) which assigns to each f [member of] [L.sup.2](G) a function in [GAMMA] which is called its Fourier transform and is denoted by f^. This function f^ is not necessarily given by the formula (9) but is defined in [L.sup.2](G) as a limit. The operator [LAMBDA] turns out to be a linear isometry, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with inverse V in the [L.sup.2] sense; this means

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

implying that f^ [disjunction]] (x) = f(x) except for a set of Haar measure 0 and often written for convenience more simply as f^ [disjunction]] ~ f. When f [member of] [L.sup.2](G) is continuous and f^ [member of] [L.sup.1]([GAMMA]), then (10) holds pointwise; and f^^(x) = f(-x) for each x in G. If f is not continuous, the representation is interpreted as holding everywhere except on a set of Haar measure 0. The related Parseval formula that for f, g [member of] [L.sup.2](G),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

can be deduced in a manner much like the classical case, as can the abstract version of the convolution theorem that for each f, g [member of] [L.sup.2](G),

(f *G g)^ = f^ g^,

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The 'shift' theorem of classical analysis also goes over to the abstract case: given f [member of] [L.sup.2](G), the Fourier transform of the 'shift' [f.sub.y](x) = f(x - y) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The integral of a character over a compact group G can be regarded as the inverse Fourier transform of the function c: G [right arrow] C which is always 1, i.e., c(x) = 1 for each x [member of] G. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

But (x, 0) = 1, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the Haar measure of G. On the other hand, if [gamma] [not equal to] 0, then there is a y [member of] G such that (y, [gamma]) [not equal to] 1. It follows (just as in classical Fourier series, when G = [T.sup.1] and (x, [gamma]) = [e.sup.2[pi]ix[gamma]) using the multiplicative property of characters (4) and the translation invariance of the integral (5) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

whence [[integral].sub.G](x, [gamma])[dm.sub.G](x) vanishes for [gamma] [not equal to] 0. Thus we get the orthogonality relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[delta].sub.[gamma],[gamma]'] = 1 if [gamma] = [gamma]' and is 0 otherwise. Now consider the Fourier transform c^ of the function c:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence, the Inversion Formula (10) and (7) imply that for each x [member of] G,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

giving the reciprocity relation

1 = [m.sub.[GAMMA]] ({0}) [m.sub.G](G) (15)

between the dual Haar measures for any compact group G and its discrete dual [GAMMA].

2.5 Annihilators, quotient groups and transversals

Let [LAMBDA] be a closed subgroup of [GAMMA]. The annihilator of [LAMBDA] is denoted by [LAMBDA] [perpendicular to] and defined by

[LAMBDA][perpendicular to] = {x [member of] G: (x, [lambda] = 1 for all [lambda] [member of] [LAMBDA]}.

It is a closed group which plays an important part in sampling theory by linking the sampling set with the spectral translates and because it is the dual ([GAMMA]/[LAMBDA])^ of the group [GAMMA]/[LAMBDA] [30, Theorem 2.1.2]. Throughout this paper, the annihilator [LAMBDA] [perpendicular to] of [LAMBDA] will be written H. A transversal [OMEGA] say is chosen for the quotient group [GAMMA]/[LAMBDA]. From now on, we shall assume that [LAMBDA] is a discrete countable subgroup of [GAMMA] such that [GAMMA]/[LAMBDA] is compact, so that [m.sub.[GAMMA]]/[LAMBDA]]([GAMMA]/[LAMBDA]) is finite. This assumption is not a serious restriction since any connected locally compact group always contains a discrete finitely generated subgroup (and so countable) such that the quotient group is compact [28, [section] 2.21]. It is clearly true in the classical Sampling Theorem, where G = R = [GAMMA], [LAMBDA] = 2WZ and [GAMMA]/[LAMBDA] [congruent] [T.sup.1]. The Haar measures of A and F/A are normalised so that the coset decomposition formula (6) holds. The Haar measures of the discrete group A and the compact group G/H are further normalised so that the inversion theorem (10) holds. Then by (15) and since [LAMBDA]^ = (H [perpendicular to])^ = G/H and since H^ = [GAMMA]/[LAMBDA], it follows that

[m.sub.[LAMBDA]]({0}) [m.sub.G/H](G/H) = [m.sub.H]({0}) [m.sub.[GAMMA]]/[LAMBDA]([GAMMA]/[LAMBDA]) = 1. (17)

The reciprocity relation (15) ensures that if the Haar measure for a compact group is chosen to be unity, the Haar measure on the discrete dual group has to be counting measure (cf. [30, p 2]). Thus, for a finite group G of order n, choosing [m.sub.G](G) = #G, the number of elements in G (or the cardinality of G), forces [m.sub.G]({0}) = 1 and the point measure of the identity [m.sub.[GAMMA]]({0}) of the dual group [GAMMA] to be [m.sub.[GAMMA]]({0}) = 1/(#G) and [m.sub.[GAMMA]([GAMMA]) = 1. For the infinite compact group [T.sup.1] with dual Z, [m.sub.Z]({0})[m.sub.T.sup.1]([T.sup.1]) = 1.

By definition, [OMEGA] consists of one and just one point from each distinct coset [[gamma]] = [LAMBDA] + [gamma], i.e., [OMEGA] [intersection]([LAMBDA] + [gamma]) consists of a single point in [OMEGA]. Thus, translates of [OMEGA] by non-zero elements in A are disjoint and the counting function [N.sub.[OMEGA]] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all [gamma], [member of] [GAMMA] and is constant on cosets, so that [??][OMEGA]([gamma]]) = Nn(7) is well defined. For each h [member of] H, the related function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is also constant on cosets. Coset decomposition will now be used to express the Haar measure of the transversal [OMEGA] in terms of the Haar measures of the compact quotient [GAMMA]/[LAMBDA] and the point {0} in [LAMBDA].

Lemma 1. The Haar measure of the transversal [OMEGA] of [GAMMA]/[LAMBDA] satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This lemma and the reciprocity relations (17) imply that for any transversal [OMEGA] of [GAMMA]/[LAMBDA], the following measure relations hold.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where H = [LAMBDA] [perpendicular to] and K is a transversal of G/H.

Since [GAMMA]/[LAMBDA] is compact, its characters [[gamma]] [right arrow] (h, [[gamma]]), h [member of] H, are orthogonal. Given any transversal [OMEGA] of [GAMMA]/[LAMBDA], the orthogonal characters of [L.sup.2]([GAMMA]/[LAMBDA]) map to functions [[eta].sub.h]: [GAMMA] [right arrow] C, where [[eta].sub.h] ([gamma]) = (h, [gamma]) X[OMEGA]([gamma]), that are an (orthogonal) basis for the space [L.sup.2]([OMEGA]), the space of functions [phi] [member of] [L.sup.2]([GAMMA]) for which [phi] vanishes almost everywhere outside [OMEGA].

Lemma 2. For distinct h, h' [member of] H, the functions [[eta].sub.h] : [GAMMA] [right arrow] [T.sup.1] satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and are an orthogonal basis of [L.sup.2] ([OMEGA]).

Proof. To prove orthogonality, it suffices to consider the simpler integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now using the coset decomposition formula,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

since [GAMMA]/[LAMBDA] is compact. That the set {[[eta].sub.h] : [OMEGA] [right arrow] C: h [member of] H} a basis for [L.sup.2]([OMEGA]) follows immediately from the characters of [GAMMA]/[LAMBDA] being a basis for [GAMMA]/[LAMBDA].

A useful by-product is the result that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The transversal [OMEGA] will not in general be compact or even bounded (see  for a discussion of this in the classical case). Fortunately, the spaces [L.sup.2]([OMEGA]) and [L.sup.2] ([GAMMA]/[LAMBDA]) are isomorphic and 'essentially' isometric. This relies on the observation that if [phi] [member of] [L.sup.2] ([OMEGA]), then for distinct [lambda], [lambda]' and for almost all [gamma] [member of] [GAMMA],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows that for almost all [gamma] [member of] [GAMMA], the function [PHI] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is constant on cosets, i.e., for any [lambda] [member of] [GAMMA], [PHI] ([gamma] + [lambda]) = [PHI]([lambda]). This means we can define a new function [??] unambiguously on the elements [[gamma]] of the transversal [OMEGA] by putting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

These functions satisfy the 'Pythagorean' property

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and allow us to pass from [L.sup.2]([OMEGA]) to [L.sup.2]([GAMMA]/[LAMBDA]) and back. It can be shown that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which gives the essential isometry referred to above (for details see ).

2.6 The Fourier transform of a sum

Let A be any subset of [GAMMA]. As we have just seen, for each [phi] : [GAMMA] + C, the sum

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

induces a function [PHI] on [GAMMA]/[LAMBDA].

Lemma 3. Let A [??] [GAMMA] with m[LAMBDA](A) < [infinity]. Suppose [phi] [member of] [L.sup.2] ([GAMMA]) and [phi] vanishes almost everywhere outside A. Then [PHI] is integrable (i.e., [PHI] [member of] [L.sup.1]([GAMMA]/[LAMBDA]) and its Fourier transform is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. To prove [PHI] integrable ([PHI] [member of] [L.sup.1]([GAMMA]/[LAMBDA])), it is enough to show that the integral below is finite. This is done by using the definition of [PHI], the triangle inequality, the coset decomposition formula and the Cauchy-Schwarz inequality, as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [PHI] [member of] [L.sup.1] ([GAMMA]/[LAMBDA]). The Fourier transform is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3 The abstract Sampling Theorem

A discussion of the abstract Sampling Theorem can also be found in . We begin by introducing the abstract version of the set [S.sub.A] of the signals in the with frequencies confined to A.

3.1 The space [S.sub.A]

Let A be a subset of [GAMMA] with positive and finite Haar measure m[GAMMA](A) (the set A corresponds to the set of frequencies of the signals being considered). The abstract analogue of the signals with frequencies in A [subset] R is the class

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

where C(G) is the set of all continuous functions f: G [right arrow] C (A does not necessarily satisfy the disjoint translates condition nor have finite Haar measure). The classical band limited condition on the signals places them in a Paley-Wiener space, but the translates condition (23) below makes A a subset of a transversal of a compact quotient group of F. While this transversal has finite measure, so that A has finite measure, A can still be unbounded (see ).

3.2 A translates condition

If the intersection of the set A and its translates A + [lambda] are null for all non-zero [lambda] [member of] [LAMBDA], i.e., if

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

then we will say that A is an almost disjoint translates set. An almost disjoint translates set is almost contained in a transversal, [OMEGA] say, of [GAMMA]/[LAMBDA], so that A [union] ([GAMMA] \ [OMEGA]) is of Haar measure zero and m[GAMMA](A) [less than or equal to] m[GAMMA]([OMEGA]). The counting function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

induces the function [[??].sub.A]([[gamma]]) on [GAMMA]/[LAMBDA] and by Lemma 3, for each h [member of] H,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Note that A is almost contained in a transversal of [GAMMA]/[LAMBDA] if and only if [N.sub.A] ([gamma]) [less than or equal to] 1 for almost all [gamma] [member of] [GAMMA]. Note, too, that the almost disjoint translates condition implies that for f [member of] [S.sub.A] (so that f^ vanishes almost everywhere outside A), and for almost all [gamma] [member of] [GAMMA],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

whence the Pythagorean property (20) holds for [N.sub.A] and [[??].sub.A].

3.3 A connection with the Borel-Cantelli Lemma

A suggestive formulation is to suppose that A [??] R and to ask how often a coset [gamma] + cZ, where c [member of] R, of a discrete subgroup cZ of R meets A. An engineering analogy is to imagine an infinite array of regularly spaced probes c units apart being used to detect material which is distributed along a line and which forms the set A.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Figure 5: An array of probes distance e part, displaced from the origin by [gamma]. The bold line represents the set A.

The set of probes can be moved along by altering the value of [gamma]. The periodicity means that the picture is unchanged if the probes are moved by an integer multiple of c, their distance apart. Thus, it is enough to consider sliding the array along up to c units (or allowing [gamma] to increase little by little from 0 to c). A precise answer to the question "How often do the probes land in A?" needs additional information but a less precise question can be answered. If the length of the material is finite (i.e., if |A| < [infinity]), then almost no cosets meet A infinitely often (i.e., [N.sub.A]([gamma]) = [infinity] for almost no [gamma]) and that if the length of the material is infinite (i.e., if |A| = [infinity]), then a set of cosets of positive measure will meet A infinitely often (i.e., the points [gamma] such that [N.sub.A]([gamma]) = [infinity] will form a set of positive length). This observation is proved for locally compact abelian groups.

Lemma 4. Let A be any Haar measurable subset of [GAMMA]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. For

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence if m[GAMMA](A) is finite, then [[??].sub.A] [member of] [L.sup.1]([GAMMA]/[LAMBDA]) and [[??].sub.A]([[gamma]]) < [infinity] for almost all [[gamma]] [member of] [GAMMA]/[LAMBDA], whence N([gamma]) < [infinity] for almost all [gamma] [member of] [GAMMA].

If m[GAMMA](A) is infinite, then [[??].sub.A] ([[gamma]]) must be infinite on a set of positive measure.

Lemma 4 is reminiscent of the pair-wise quasi-independent extension [6, 15, 31] of the Borel-Cantelli Lemma.

Lemma 5. Let [E.sub.n], n [member of] N be events in a probability space with probability P. Let N (E) be the number (possibly infinite) of times E occurs. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It can be shown that when the events [E.sub.m] are pair-wise independent (i.e., when P([E.sub.m] [intersection] [E.sub.n]) = P([E.sub.m]) P([E.sub.n]) for m [not equal to] n), the probability of the events occurring infinitely often is 1 when the sum diverges. The original Borel-Cantelli Lemma requires total independence of the events in the case of divergence, so the pair-wise versions represent an enormous improvement and deserve to be better known.

3.4 The abstract WKS Theorem

The natural step of extending the WKS theorem to general locally compact abelian groups, first taken by I. Kluvanek , was extended slightly in  (a fuller more comprehensive mathematical account is in ). The starting point for placing the WKS Theorem in an abstract setting is to replace the time domain, represented by the real line R, by a locally compact abelian group G, written additively. Thus, instead of a function f: R [right arrow] C being reconstructed from values at regularly spaced points k/2W in R, a function f: G [right arrow] C on a locally compact abelian group G is reconstructed from samples f(h) taken at points h in a discrete subgroup H of G.

There are a number of ways of proving the Kluvanek's abstract version of the WKS Theorem and his method of proof , which uses the traditional approach that the functions r/h form an orthogonal basis for [L.sup.2] ([OMEGA]), will be followed (other proofs are given in [2, 11]).

Theorem 1. Suppose that the dual F of the locally compact abelian group G has a discrete subgroup A with [GAMMA]/[LAMBDA] compact. If A [??] [LAMBDA] is a subset of a transversal [OMEGA] of [GAMMA]/[LAMBDA] except for a null set, then m[GAMMA](A) < [infinity] and each f [member of] [S.sub.A] has a representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where H = [[LAMBDA].sup.[perpendicular to]] and the convergence is in the [L.sup.2] sense and uniform. Moreover,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. By hypothesis, the quotient group [GAMMA]/[LAMBDA] has a transversal [OMEGA] say which almost contains A, so that m[GAMMA](A) [less than or equal to] m[GAMMA]([OMEGA]) < [infinity]. By definition of [S.sub.A] (22) and the finiteness of m[GAMMA](A), for each f [member of] [S.sub.A], f^ [member of] [L.sup.2](A) [intersection] [L.sup.1](A) [??] [L.sup.2]([OMEGA]) and so by Lemma 2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

whence taking the inverse Fourier transform,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by the continuity of f. Choosing x = h' [member of] H, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by (19), so that

[a.sub.h] = f (-h)/m[GAMMA]([OMEGA])

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

since H = -H. The sum

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by Plancherel's formula (11).

The theorem can also be established using the last sum as a starting point [21, Theorem 10.1, Chapter 10]. For, just as in the classical theory, one can deduce in turn analogues of Parseval's theorem and the convolution theorem, which implies that for [psi], [phi] [member of] [L.sup.2]([GAMMA]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

By putting f^ = [psi] and XA = [phi], and observing that the translates condition (23) implies the tautology

f^ = f^ XA,

we get by the continuity of f that

f^[disjunction] = f = 1/m[GAMMA]([OMEGA]) f * H [X.sup.[disjunction].sub.A.

4 The converse to the abstract Sampling Theorem

In 1959 Belyaev  and Lloyd  extended the WKS theorem to the almost sure reconstruction of stochastic processes from linear combinations of samples and obtained the converse (see [21, Chapter 9] for more details). A converse of the multi-band result for the classical case of continuous, square-integrable functions is given in . The argument relies on finding a suitable function from the class [S.sub.A] (22). This is not so straightforward for functions on locally compact abelian groups, so a direct proof is sketched (see  also). In this, the representation (24) for f in Kluvanek's theorem implies that A satisfies the almost disjoint translates condition with respect to A (23).

Theorem 2. Suppose that the dual F of the locally compact abelian group G has a discrete subgroup [LAMBDA] with [GAMMA]/[LAMBDA] compact. Suppose the Haar measure [m.sub.[GAMMA]](A) of A [subset] [GAMMA] is finite and that for each f [member of] [S.sub.A],

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

where the convergence is in the [L.sup.2] sense and uniform. Then for each non-zero [lambda] [member of] [LAMBDA], [m.sub.[GAMMA]](A [intersection] (A + [lambda])) = 0.

Proof. Since [m.sub.[GAMMA]](A) < [infinity], [chi square]A [member of] [S.sub.A] [subset] [GAMMA], whence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Taking the Fourier transform,

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

Now let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (26)

Then by the Inversion Theorem for [[??].sub.A]: [GAMMA]/[LAMBDA] [right arrow] C and by Lemma 3,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

since H = -H, H is the annihilator of [LAMBDA] (16) and [m.sub.[GAMMA]]([OMEGA]) = [m.sub.[LAMBDA]]({0})/[M.sub.H]({0}) by (18). Clearly, XA(a) = 1 for a [member of] A, whence (25) gives for almost all a [member of] A,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by (26), i.e., for almost all a [member of] A,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence for [lambda] [not equal] 0, XA (a + [lambda]) = 0 for almost all a [member of] A. Thus, for almost all a [member of] A, a + [lambda] [??] A for [lambda] [not equal to] 0 and the result follows.

The equivalence follows on putting Theorems 1 and 2 together.

Theorem 3. Let G be a locally compact abelian group with dual F having a discrete subgroup [LAMBDA] such that [GAMMA]/[LAMBDA] is compact. Let H = [LAMBDA] [perpendicular to]. The following are equivalent.

1. [m.sub.[GAMMA]] (A [intersection] (A + [lambda] = 0 for all non-zero [lambda] [member of] [LAMBDA].

2. [m.sub.[GAMMA]] (A) < [infinity] and for each f [member of] [S.sub.A],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the convergence is in the [L.sup.2] sense and uniform.

The reconstruction process of real signals from samples can be derived from the norm [parallel]f[parallel]G, the energy of f, being equal to the norm [parallel]f[parallel]H, the energy of the samples f(h), modulo a constant. This gives another equivalence result.

Theorem 4. The following are equivalent under the same hypotheses as Theorem 3.

1. [m.sub.[LAMBDA]](A [intersection] (A + [lambda])) = 0 for all non-zero [lambda] [member of] [LAMBDA]

2. [m.sub.[LAMBDA]](A) < [infinity] and for each f [member of] [S.sub.A],

[parallel] f [parallel] H = [m.sub.[GAMMA][([OMEGA]).sup.1/2[parallel] f [parallel]G.

When the second hypothesis holds, the energy of the 'sampled' signal is essentially the same (modulo a multiplicative constant) as the original signal and the two signals contain the same information (see [section]10.5.2 in ).

5 Applications

While fundamental in the theory of communication, in practice the formula given by the Sampling Theorem is not used directly, as the convergence is slow and is not absolute. In signal processing, the so-called 'sample and hold' technique is widely used (see  and references therein) instead of the sampling formula. However, the theorem can be used to estimate reconstruction errors arising in this technique  and it provides a way to avoid spectral overlaps for certain types of signal .

A wide range of applications of the theorem is to be found in Rowland Higgins' fascinating expository account , which includes a discussion of a formula due to Cauchy and a sampling theorem for 'sequency-limited' signals using Walsh functions in their ro1e of characters for dyadic groups. Some other examples are now given to illustrate the usefulness of the abstract approach.

5.1 The Whittaker--Kotel'nikov--Shannon theorem

In the WKS theorem, G = [LAMBDA] = R, [m.sub.R] is the usual Lebesgue measure or essentially 'length', A = [OMEGA] = [--W,W) and (x,y) = [e.sup.2[pi]ixy, [LAMBDA] = 2WZ, with counting measure. The sampling set from which the signal can be reconstructed is H = Z/(2W) = [(2WZ).sup.[perpendicular to], the annihilator of 2WZ, with counting measure. The set [0, 1/(2W)) is a transversal K of R/(Z/(2W)) and

[[m.sub.R]/(2WZ).sup.(R/(2WZ))] [m.sub.2WZ]({0}) = |[--W,W)| =2W

and

[[m.sub.R]/(Z/(2W)).sup.(R/(Z/(2W))) = |[0,1/(2W))| = 1/(2W).

The formula (1) follows from Theorem 1, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For the multi-band generalisation of the Whittaker-Kotel'nikov-Shannon theorem, where the interval [-W, W] is replaced by an almost disjoint translates set, see [3, 10].

5.2 The discrete Whittaker--Kotel'nikov--Shannon theorem

A 1-periodic function f: R [right arrow] C can be considered to be a function on the half open unit interval [0, 1) or on the quotient group R/Z by defining/([t]) = f(t), where [t] = t + R, the coset represented by t [member of] R. Since R/Z is isomorphic to the unit circle [T.sup.1], a 1-periodic function f (or indeed any periodic function) can be considered as a function on the circle. The Fourier transform f^ of such periodic functions is a Fourier series, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where the coefficients [a.sub.k] = f^(k). In the discrete version of the Whittaker--Kotel'nikov--Shannon theorem, periodic functions with bounded Fourier transforms, i.e., with bounded frequencies, are considered. Such functions are trigonometric polynomials of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where t [member of] [0, 1) and [c.sub.j] = f^(j) [member of] C. The discrete Whittaker--Kotel'nikov-Shannon theorem states that such functions can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where A = [OMEGA] = {0, [+ or -] 1, ... , [+ or -]N}, [m.sub[GAMMA]]([OMEGA]) = 2N + 1, G = (0, 1] mod 1 [congruent to] [T.sup.1], [GAMMA] = Z (a discrete infinite group but for k [member of] N with Z/(kZ) [congruent ot] [Z.sub.k] compact), [LAMBDA] = (2N + 1)Z, [GAMMA]/[LAMBDA] = Z/((2N + 1)Z) [congruent to] [Z.sub.2N + 1], and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For details see ; Rowland Higgins has kindly reminded me that the discrete sampling theorem is due to Cauchy . For the discrete analogue of the general sampling theorem using disjoint translate sets, see [8, 10].

5.3 Sampling in the plane

Let G = [R.sup.2], so that Haar measure is planar Lebesgue measure or 'area'. Consider a continuous planar function f which is square integrable (f [member of] [L.sup.2]([R.sup.2])) with Fourier transform f^ vanishing outside the set

A = {(x,y): max{|x [+ or -] 10|, |y|} < 1/2} [union] {(x,y): max{|x|, |y [+ or -] 10|} < 1/2},

consisting of four unit squares lying on a coordinate axis and centred 10 units from the origin. Then the area ([m.sub.R.sup.2](A)) of A is 4 and it can be seen that for any (j, k) [member of] [Z.sup.2],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus, [LAMBDA] = 3[Z.sup.2] and has annihilator H = [Z.sup.2]/3. A transversal for [GAMMA]/[LAMBDA] is [OMEGA] = 3[[0, 1).sup.2], with [m.sub.[GAMMA]]([OMEGA]) = (3[[0, 1).sup.2]) = 9, whence for each (x,y) [member of] [R.sup.2],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[X.sup.[disjunction].sub.A] (x,y) = 1/[[pi].sup.2]xy

Note that the area of A is 4 < 9, the area of the transversal [OMEGA] = [3.sup.2][[0, 1).sup.2] for [R.sup.2]/3[Z.sup.2]. The two-dimensional sampling theorem has the simpler reconstruction kernel

[S.sup.[disjunction].sub.A] (x,y) = 4sin21[pi]x sin21[pi]y/441[pi].sup.2]xy

but this is paid for by the much higher sampling rate of 441 (for more details and other reconstructions, see ).

5.4 Scanning

During the discussion following the delivery of this paper, Hans Feichtinger pointed out that medical scanning of the body involves sampling horizontally and through 360[degrees], corresponding to the product group R x [T.sup.1]. The corresponding sampling formula is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In practice the different Radon transform is used (see [7, 16, 29]).

ACKNOWLEDGEMENTS

Much of this work was done with Michael Beaty. I am grateful to him and to Paul Butzer, Simon Eveson, Hans Feichtinger, and Rowland Higgins for their continued interest and support. Thanks are also due to the Organising Committee of Sampta05 for the very successful and enjoyable meeting at the Ondokuz Mayis University, Samsun and to the referees' careful checking and perceptive comments, which have greatly improved the article.

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M. M. Dodson

Department of Mathematics, University of York

Heslington, York YO10 5DD, UK

mmd1@york.ac.uk
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