# Sampling in shift-invariant spaces with Gaussian generator.

Abstract

We extend Unser's theory of sampling in shift-invariant spaces to the case of a Gaussian generator previously excluded because of violating the partition of unity condition. An unconditional sampling (or interpolation) theorem for these spaces is stated and proved. As an application the interpolation expansion of the dual function is given. Then, for a new class of function spaces that we call exponential Sobolev spaces, a sampling theorem is stated and proved. Exponential Sobolev spaces are more general than shift-invariant spaces but reconstruction of sampled functions takes place in a shift-invariant space with Gaussian generator. Finally it is noted that, despite a striking similarity between two arising interpolating functions, these are not the image of each other under a connected Gauss transform.

200 AMS Mathematics Subject Classification--94A20, 41A05

Key words and phrases : Gaussian function, Jacobi theta-functions, generator, dual function, interpolating function, interpolation theorem, Gauss transform, exponential Sobolev space, sampling theorem

1 Introduction

Unser  extends Shannon's  model of sampling by replacing the space of bandlimited functions by more general function spaces of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [phi] [member of] [L.sup.2](R) is the so-called generating function (or generator ) and [[??].sup.2] is the space of square-summable complex valued sequences. V([phi]) is a closed subspace of [L.sup.2](R) and shift-invariant in the sense that f(.-k) [member of] V([phi]) whenever f [member of] V([phi]) and k [member of] Z. The following additional conditions are imposed on the generator [phi]; we refer to  concerning the relevance of these conditions:

(1) The family of functions {[[phi].sub.n] = [phi](.-n); n [member of] Z} forms a Riesz basis,

(2) [phi] satisfies the partition of unity condition (PUC)

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

It is the PUC (1) which precludes the Gaussian probability density function of standard deviation 1/[beta], [beta] > 0,

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

from being an admissable generator, (1) see Figure 1(a).

[FIGURE 1 OMITTED]

On the other hand, as shown by Figure 1(b), in case [phi] = [[phi].sub.0] the sum in (1) becomes close to 1 as the bandwidth parameter [beta] decreases. Indeed, by the Poisson summation formula (see [2, 10]) we know that

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

and because of exponential decay of the Gaussian function exp[-[[xi].sup.2]/(2[[beta].sup.2])] outside of the interval [-[pi][beta], [pi][beta]] the terms in (3) with index n [not equal to] 0 will contribute to almost nothing provided that [beta] is sufficiently small. Then the PUC will be met with high accuracy and the Gaussian function (2) still might be a suitable generator.

The goal of the paper is to substantiate the usefulness of Gaussian functions as generators for shift-invariant spaces leading to new sampling theorems. We are only concerned with regular sampling; the use of Gaussian generators in nonuniform sampling  is not investigated.

The paper is organized as follows: In Section 2 we state and prove an interpolation theorem for a shift-invariant subspace [V.sub.[lambda]] of [L.sup.2](R) with Gaussian generator. As an application, in Section 3 we compute the interpolation expansion of the dual function [[phi].sub.dual] [member of] [V.sub.[lambda]]. In Section 4 the exponential Sobolev spaces are introduced, and a sampling theorem is stated and proved.

The paper has been presented in part on the International Workshop SampTA 05, Samsun, July 10-15, 2005.

2 Interpolation Theorem

Similarly to the space [V.sub.T]([phi]) in  we introduce the shift-invariant space [V.sub.[lambda]] [??] [L.sup.2](R) defined by

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

where [phi]0 is the Gaussian function (2) and [lambda] is an arbitrary positive number having the meaning of the sampling interval later on.

The family of functions {[phi]0(.-n[lambda]);n [member of] Z} again forms a Riesz basis of [V.sub.[lambda]] and the PUC, now reading

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

is still unsatisfied (yet holding with high accuracy as [lambda] [right arrow] 0, [beta] > 0 held constant). Since [V.sub.[lambda], may also be defined as the closed linear span in [L.sup.2](R) of {[phi]0(.-n[lambda]); n [member of] Z}, it forms a Hilbert space with respect to the inner product ([f.sub.1], [f.sub.2]) = [f.sub.R][f.sub.1](x)[f.sub.2](x) dx of [L.sup.2](R). [V.sub.[lambda]] coincides with the space of the same notation in  if the parameter [alpha] controlling time-duration (see also Section 4) tends to [infinity]. The dual function [[phi].sub.dual] (see , denoted by [f.sub.0] in ) is the uniquely determined element of [V.sub.[lambda]] satisfying the biorthogonality condition

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

In  the Fourier transform of the dual function in case [alpha] < [infinity] has been computed whence we obtain as [alpha] [right arrow] [infinity] that

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

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

and the Jacobi theta-function [[upsilon].sub.3](*, [theta]) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

(see ). [V.sub.[lambda]] forms a reproducing kernel Hilbert space (RKHS) with reproducing kernel

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Indeed, (1) because of the rapid decay of [phi]0 the function y [right arrow] q(x, y) is an element of [V.sub.[lambda]] for any x [member of] R held constant, (2) since every f [member of] [V.sub.[lambda]], has an expansion [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] , it is readily seen by biorthogonality (6) that

<f, q (x,*)> = f(x)[for all]x [member of] R. (10)

This observation is due to Walter  who used the RKHS property of certain wavelet subspaces for the derivation of sampling theorems. The crucial condition in  on the scaling function [phi] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

Moreover, it is assumed that [phi] forms an orthogonal basis {[phi](x - n)} of the wavelet subspace, [V.sub.0]. The latter assumption is also made in Walter and Shen's book , but the proof of the next theorem follows closely the exposition in [12, pp. 189-191]. An extension of the RKHS setting to the biorthogonal case has been indicated in .

The following theorem holds unconditionally for any [lambda] > 0 (in contrast to Theorem 3, see below), which is why we call it an interpolation theorem. We use the Jacobi theta-function [[upsilon].sub.1](*,[theta]), which for an arbitrary parameter [theta] [member of] C, [??]([theta]) > 0 and variable z [member of] C is given by

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

where q = [e.sup.i[pi][theta]] .

Theorem 1 (Interpolation Theorem) Let [lambda] > 0. For any f [member of] [V.sub.[lambda]] it holds that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

where the function [[phi].sub.int] [member of] [V.sub.[lambda]] is given by

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

with [theta] as in (8). The series converges in [L.sup.2](R) and uniformly on R.

Proof. Since the family of functions {[phi] dual (* - n[lambda]); n [member of] Z} is biorthogonal to {[[phi].sub.0](* - n[lambda]);n E Z} it also forms a Riesz basis of [V.sub.[lambda] .

We now show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is another Riesz basis of [[??].sub.[lambda]. The Fourier transform maps [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] onto [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [LAMBDA] as in (8). We define the operator I on [V.sub.[lambda]] by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In the Fourier domain the corresponding operator [??] maps P([xi][phi]dual to P([phi])[PHI]([xi])dual, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Here we have used the Fourier series representation of the theta-function (9),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where q = [e.sup.i[pi][pi]] . There exist constants 0 < A [greater than equal to] B < [infinity] such that A [less than equal to] [PHI][xi]dual [less than or equal to] = B [for all][xi] [member of] [0, [LAMBDA]] (implying that condition (11) is satisfied). Hence, [??]: [[??].sub.[lambda]] [right arrow] [[??].sub.[lambda]] is a bounded operator. Likewise, the operator [??] on [[??].sub.[lambda]] defined by [??](P([xi])[phi]dual) = (P([xi])/[PHI]([xi]))[PHI] is a bounded operator from [[??].sub.[lambda]] into itself. Thus, the operator [??] maps [[??].sub.[lambda]] onto itself and has the bounded inverse [[??].sup.-1] = [??]. Since for the operator I: [[??].sub.[lambda]] [right arrow] [[??].sub.[lambda]] it holds the same, I({[phi]dual(. - n[lambda]); n [member of] Z}) = {q(n[lambda],.); n [member of] Z is a Riesz basis of [[??].sub.[lambda]].

Consequently, there exists a family of functions {[S.sub.n]; n [member of] Z} [??] [[V].sub.[lambda]] biorthogonal to {q(n[lambda],.); n [member of] Z}. It again forms a Riesz basis of [V.sub.[lambda]] and is uniquely determined . Since q(n[lambda] y) = q(0, y - n[lambda]), it holds that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because of the uniqueness of the biorthogonal system, we have sin(x) = so(x - m[lamdba]). Conforming with  we use the notation so = [phi]int and call [phi]int an interpolating function.

As a result, any function f [member of] [V.sub.[lambda]] has an expansion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next, we calculate (Pint. Taking specifically f = [phi]0 and performing Fourier transformation on both sides of (13), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [PHI]([xi] is as in (15). Since [PHI]([xi]) [not equal to] 0, we may divide and get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Inversion of the Fourier transform yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we have used the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which has been derived in  for an arbitrary parameter r = int, t > 0. Application of Jacobi's imaginary transformation [13, 6],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

finally results in the representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By means of the Cauchy-Schwartz inequality we infer from (10) that for any v [member of] [V.sub.[lambda] it holds that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For ||q(x,*)|| we get the estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with a constant [C.sub.1] < [infinity] not depending on x. So, we have

|V(X)| [less than or equal to] [C.sub.2]||v||[for all]x [member of] R,

with a constant [C.sub.2] < [infinity] not depending on x and v. Substituting v by any difference of partial sums [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] shows that [f.sub.N] [right arrow] f not only in [L.sup.2] (R) but also uniformly on R as N [right arrow] [infinity]. This concludes the proof of Theorem 1.

We remark that the interpolating function [phi]int has the property

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In Figure 2, (flint for parameters [beta] = 10.0, [lambda] = 0.1 is depicted.

[FIGURE 2 OMITTED]

3 Application: Expansion of the Dual Function

The computation of the dual function, [phi]dual, by direct inversion of the Fourier transform (7) is difficult. It becomes considerably simpler when restricting the time variable x to sampling instants n[lambda], n [member of] Z. Since [phi]dual [member of] [V.sub.[lambda]] by Theorem 1 we then obtain the dual function for any x [member of] [R] by its interpolation expansion.

Theorem 2 Let [lambda] > 0. The dual function [phi]dual [member of] [V.sub.[lambda]] has the expansion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Inversion of the Fourier transform (7) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We compute the latter integral for sampling instants x = n[lambda], n [member of] Z. By means of the identity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which holds for an arbitrary parameter [tau], we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Since [[upsilon].sub.3](z,[tau]) = [[upsilon].sub.0] (z + 1/2, [tau]) , we obtain for the integral on the right-hand side of (19) (denoted by [J.sub.0](n, [tau])) after substitution of s by s - 1/2 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the evaluation of the integral [I.sub.0] (n, [tau]) we use the theorem of residues . We integrate the function [w.sub.n](Z) = exp(2[pi]nz) [[upsilon].sub.0](z, 2[tau])/[[upsilon].sub.0] (Z, [tau]) in the complex plane counterclockwise along the contour formed by the rectangle whose corners are -1/2, 1/2, 1/2 + M * 2[tau], -1/2 + M * 2[tau] where M is an arbitrary positive integer. Because [W.sub.n](Z) = [W.sub.n](z + 1) [for all]z [member of] C, the integrals along the sides parallel to the imaginary axis cancel each other. Since z [right arrow] [upsilon].sub.0] (z,r) has simple zeros in the points [z.sub.m] = (m + 1/2) [tau], m = 0,1, ..., 2M - 1 (see ), [w.sub.n](Z) is an analytic function in that domain with the exception of the points [z.sub.m] where it has simple poles. Hence, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Application of the identity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to numerator and denominator of the integrand function in (21) (for parameters [tau] and 27, resp.) shows that [W.sub.n] (S + M * 2[tau]) [right arrow] 0 as M [right arrow] [infinity] uniformly for s [member of] [-1/2, 1/2]. Consequently, [I.sub.M] (n, [tau]) [right arrow] 0 as M [right arrow] [infinity] and, therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, we need to compute the residues

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the identity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]  we infer that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Next, [[upsilon].sub.0]([z.sub.m],2[tau]) = [[upsilon].sub.0] ([tau]/2 + m/2 * 2[tau], 2[tau]). We treat the two cases m = 2k and m = 2k + 1, k = 0, 1, ... separately. In the case m = 2k we obtain by one of the earlier mentioned identities for the theta-funtion [v.sub.0] that [[upsilon].sub.0]([z.sub.m],2[tau]) = [(-1).sup.k][[upsilon].sub.0]([tau]/2, 2[tau])[q.sup.-1/2m(m+1)]. In the case m = 2k + 1 we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] infinite series for the two values shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, in any case we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Putting all together, we obtain after a simple calculation

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Collecting all, we obtain [phi]dual (n[lambda]) for arbitrary n [member of] Z. Since [phi]dual is an even function, ([phi]dual(n[lambda]) = ([phi]dual|n|[lambda]), n = -1, -2, ... Application of Jacobi's imiginary transformation [13, 6] to the two theta-functions occuring in (22) yields the result; the arising Jacobi theta-function [[upsilon].sub.2] is as in . This concludes the proof of Theorem 2.

In Figure 3, [phi]dual for parameters [beta] = 10.0, [lambda] = 0.1 is depicted.

[FIGURE 3 OMITTED]

(A closer look at the curve reveals that the zeros [z.sub.n] of ([phi]dual are irregularly spaced in the neighbourhood of the origin soon becoming almost equidistant as n [right arrow] [+ or -] [infinity].)

4 Sampling Theorem

The sampling theorem for time-frequency localized signals [4, Theorem 6] becomes a sampling theorem for a new class of function spaces related to shift-invariant spaces with Gaussian generator if the parameter [alpha] (see below) controlling time-duration tends to [tends]. This theorem will be stated and proved in the present section.

The time-frequency localization operator (TFLO) [P.sub.[gamma].sub.[delta]) : [L.sup.2](R) [right arrow] [L.sup.2](R) used in  is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [alpha], [beta] are arbitrary positive numbers satisfying the condition [alpha][beta] > 1, [delta] is defined by coth [delta] = [alpha][beta], and [lambda] = [square root of alpha/beta]

For s [member of] R, s > 0 we define the Gauss transform [G.sub.s] : [L.sup.2] [right arrow] [L.sup.2] (R) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e., as the convolution with a Gaussian probability density function of variance s. We note that the family of operators {[G.sub.s]; s > 0} forms a semigroup, as does {[P.sup.[gamma].sub.[delta]] > 0}, [gamma] > 0 held constant . For any f [member of] [L.sup.2](R) it holds that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

with convergence in [L.sup.2] (R) and uniformly on any compact subset of R. Moreover, in the case of [alpha] [right arrow] [infinity] we obtain for the norm. ||*||[gamma][delta] in the so-called localization space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the TFLO (23) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here we have used that the marginal density of the Wigner distribution is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (see, e.g., ). Thus, when [alpha] [right arrow] [infinity] the localization space [P.sub.[gamma][delta]] (which is a phase space concept, see ) collapses to a function space where time and frequency behaviour of elements are treated separately again. We shall call that new function space an exponential Sobolev space because of its resemblance to classical Sobolev spaces where polynomial weights rather than exponential ones are used; see  for that and generalizations.

Definition 1 The exponential Sobolev space of order s [member of] R, s > 0 is the function space

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with inner product [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The next theorem will show that <.,.>s is indeed an inner product.

Proposition 1 For any s [member of] R, s > 0 the exponential Sobolev space [W.sup.2,exp.sub.s] is an RKHS. Gs is an isometry from [L.sup.2](R) onto [W.sup.2,exp.sub.s], i.e., [G.sub.s] ([L.sup.2](R)) = [W.sup.2,exp.sub.s] and for [g.sub.i] = [G.sub.s][f.sub.i], [f.sub.i] [member of] [L.sup.2](R), i = 1,2 it holds that

<g1,g2>s = [f.sub.1],[f.sub.2] (25)

The reproducing kernel is K(x, y) = G2s(x - y).

Proof. In the frequency domain the Gauss transform g = [G.sub.s]f where f [member of] [L.sup.2](R) reads as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

If [g.sub.i] = [G.sub.s],[f.sub.i],[f.sub.i] [member of] i = 1,2, it is readily seen that [<g1,g2>.sub.s] = <[??],[??]> = <[f.sub.1],[f.sub.2]>, proving (25). Consequently, if g = [G.sub.s]f, f [member of] [L.sup.2](R), we have ||g||s = ||f|| < [infinity]. Thus, [G.sub.s] ([L.sup.2](R)) [??] [W.sup.2,exp.sub.s]. Conversely, if g [member of] [W.sup.2,exp.sub.s], then the function f defined by [W.sup.2,exp.sub.s] is in [L.sup.2](R). Since the latter equation is equivalent to (26), it follows that g = [G.sub.s]f with f [member of] [L.sup.2](R). Thus, [W.sup.2,exp.sub.s] [??] [G.sub.s]([L.sup.2](R)). As a result, [G.sub.s]([L.sup.2](R)) = [W.sup.2,exp.sub.s]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because of the isometry relation (25), [W.sup.2,exp.sub.s] endowed with the inner product (*,*)s becomes a Hilbert space as is [L.sup.2](R). For any x [member of] R we define the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have K(x, *) [member of] [W.sup.2,exp.sub.s] exp. Furthermore, for any function g = [G.sub.s]f, f [member of] [L.sup.2](R) we obtain by means of (25) that

[W.sup.2,exp.sub.s]

Thus, K(x, y) is a reproducing kernel and 14;2'exp is an RKHS. This concludes the proof of Proposition 1. []

Because of (24), the variance parameter

s = s([beta]) = [(1/beta).sup.2]

will be the proper setting in our context. The Gaussian probability density function of standard deviation [square root of 2]/[beta]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an element of [W.sup.2,exp.sub.s]. It is the image of [phi]0 (the Gaussian probability density function of standard deviation 1/[beta]) under the Gauss transform [G.sub.s([beta])].

Definition 2 Let [lambda] > 0, [beta] > 0. The shift-invariant space [V.sub.[lambda]] is the closed linear span in [W.sup.2,exp.sub.s] of the family of functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[V.sub.[lambda]] can also be described as the image under [G.sub.s([beta])] of the shift-invariant subspace [V.sub.[lambda]] (see (4)) of [L.sup.2](R). Actually, [V.sub.[lambda]] itself may be viewed as shift-invariant subspace of [L.sub.2] (R), as shown by the following theorem.

Proposition 2 On [V.sub.[lambda]] [??] [W.sup.2,exp.sub.s] we have norm equivalence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Hence, [V.sub.[lambda]] is equivalent to the closed linear span of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in [L.sup.2](R).

Proof. We define [U.sub.[lambda] as the linear span of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Any function u [member of] [U.sub.[lambda] has the representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with corresponding Fourier transform

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [lambda] is as in (8) and p([xi]) is a Fourier polynomial of period 1. A computation yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with constants [c.sub.1] > 0, [c.sub.2] > 0 and weight functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [tau] as in (8). Since for any parameter [tau] = i[pi]t with t > 0, [xi] [right arrow] [[upsilon].sub.3] ([xi], [tau]) is a positive and bounded continuous function on the interval [0, 1], there exist constants 0 < c [less than or equal to] C < [infinity] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e., ||.||,||.||s([beta]) are equivalent norms on [u.sub.[lambda] (thus, by continuity, on [V.sub.[lambda]]). Therefore, the closure of/2~ with respect to any of the two norms will result in equivalent spaces. This concludes the proof of Proposition 2.

Hence, Theorem 1 applies and any function g [member of] [v.sub.[lambda]] can be perfectly reconstructed from sample values g(n[lambda]), n [member of] Z by the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with parameter [tau]' as given by the right-hand side of the second equation in (8) after the substitution of [beta] by [beta]/[square root of 2] (the reciprocal standard deviations of [phi]0 and [[PHI].sub.0], resp.). Since 2[tau]' = [tau] where T is again as in (8), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

The reconstruction formula (27) will now be extended to functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] resulting in an approximation [??] [member of] [v.sub.[lambda] rather than in g itself.

Theorem 3 (Sampling Theorem) Let [lambda] > 0, [beta] > 0 and let [[PHI].sub.int] [member of] [V.sub.[lambda] [??] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the interpolating function (28). Then for any g [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [??] defined by the series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

is in [V.sub.[lambda]]. When the sampling interval A is sufficiently small,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

then the error [g(x) - [??](x)| decays exponentially to 0 as [lambda] [right arrow] 0 (uniformly in x [member of R). Moreover, on the real line it holds that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with high accuracy.

Proof. Theorem 3 is a corollary of the sampling theorem [4, Theorem 6] if the time parameter, [alpha], tends to [infinity]. Then, as already seen above, the localization space [P.sup.[lambda].sub.[delta]] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the generalized translation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

becomes an ordinary translation [g.sub.0] [right arrow] [g.sub.0] (* - n[lambda]), and the so-called sampling function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

because of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

tends to the interpolating function [[PHI].sub.int] of (28) as [alpha] [right arrow] [infinity]. Moreover, the reconstruction space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of [4, Definition 4] is converted into the space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [??] as given by (29) becomes an element of [V.sub.[lambda].

We estimate the error [A.sub.[lambda]] = |g(x) - [??](x)] (which is type of an aliasing error) by means of the inequality [4, ineq. (61)]. It continues to be correct as [alpha] [right arrow] and becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

where, now, [sigma] = [beta]/[square root of 2], and [c.sub.1] > 0, [c.sub.0] = ([square root of 24] - 4) [approximately equal to] 1.11 are the same constants as in . By the same reasoning as in , it is inferred that the exponential expression on the right-hand side of (33) becomes small if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is certainly the case when [lambda] [less than or equal to] [[beta].sup.-1]. Eventually the error [A.sub.[lambda]](x) decays exponentially to 0 as [lambda] [right arrow] 0. The proof of (31) is same as in . This concludes the proof of Theorem 3. []

In Figure 4, [PHI]int for parameters [beta] = 10.0, [lambda] = [[lambda].sub.0] = 0.1 is depicted.

[FIGURE 4 OMITTED]

Remark 1. Besides the preceding "method of descent" (because of subsequent elimination of the time constraint in the original sampling theorem [4, Theorem 6]) it is also possible to recast the whole proof of the original sampling theorem in the setting of the RKHS [lambda] (instead of the RKHS [P.sup.[gamma].sub.[delta]) resulting in a