# On the eigenfunctions of the finite Hankel transform.

Abstract

In this paper, we study the properties of the eigenfunctions of the finite Hankel transform. We deduce a sampling series in terms of these functions for Hankel-band-limited signals and derive bounds for the truncation error of the sampling series.

Key words and phrases : Reproducing kernel space, Kramer sampling theorem, finite Hankel transform.

2000 AMS Mathematics Subject Classification--41A05, 65D05, 65D20.

1 Introduction

The classical Whittaker-Kotel'nikov-Shannon (WKS) sampling theorem says that any function f from the Paley-Wiener space [PW.sub.[OMEGA]], i.e. any f [member of] [L.sup.2](-[infinity], [infinity]) representable through its Fourier transform F[f] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

can be reconstructed from its equally spaced samples,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The series in the right-hand side converges both in the [L.sub.2]-norm and uniformly on (-[infinity], [infinity]). This follows from the fact that F[f] [member of] [L.sup.2]([OMEGA], [OMEGA]) and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Another sampling formula for f(t), which was invented and studied in [30, 31], is based on the eigenfunctions [[psi].sub.l]([OMEGA], x) = [[psi].sub.l] (x) of the Finite (Truncated)

Fourier Transform [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The eigenfunctions

[[psi].sub.l](x) are first defined inside the interval [I.sub.[OMEGA]] = [-[OMEGA], [OMEGA]] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

and are then continued analytically to the entire real axis according to (2). The associated TFT eigenvalues [[mu].sub.l] = [[mu].sub.l]([OMEGA]) are ordered by ma[G.sub.[OMEGA]]itude, [square root 2 [pi]] |[[mu].sub.0]| > |[[mu].sub.1]| > ... > 0.

Due to their unusual and interesting features, the TFT eigenfunctions, also known as 'prolates', have long attracted attention; we refer the reader to the classical works by Slepian, Pollak, and Landau [27, 16, 17, 18], as well as the papers [30, 31, 34] and the literature cited therein. This list of references is far from complete. The properties of prolates [[psi].sub.l] are widely discussed in the literature. Nevertheless we shall proceed with a short survey of those properties which are related to the fact that prolates form an orthonormal basis in a reproducing kernel space [5]. In such spaces Kramer's generalization of the WKS sampling theorem is valid. Kramer's theorem expands the variety of functions that can be recovered from their sampled values by means of orthogonal sampling formulae, and also establishes general conditions allowing such a recovery (see e.g. [14, 12, 35, 8, 7, 23]).

The essence of the general theory is summarized in a recent paper [36], where an algorithm to generate systems of functions possessing the above--mentioned properties of prolates is provided. We shall illustrate the general description of [36] with the example of the Finite (Truncated) Hankel Transform (THT) and its eigenfunctions. These functions are extensively used now in various areas that range from applied physics to computer science. The number of publications in which they are employed is so large that any comprehensive survey of modern applications would be a very complicated task going far beyond the present study. Here we only note that historically the THT eigenfunctions were introduced almost simultaneously in three pioneering works [26, 9, 33], each being a source of many subsequent publications on numerical optical analysis, such as both forward and inverse analysis of high numerical aperture focusing systems, calculation of mode patterns and losses in optical resonators with spherical mirrors. Among the most recent papers see, e. g. [25, 6, 3].

In the present work we pay special attention to the bounds of the error induced by the sampling series truncation. In [30, 31, 211 22] the truncation error estimates were obtained for particular classes of band--limited functions and some illustrative examples were given. In what follows we shall be concerned with another class of band-limited functions, namely with the range of the operator [G.sub.[OMEGA]] := (1/2 [pi])[F.sup.*.sub.[OMEGA]] o [F.sub.[OMEGA]], i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

One can also interpret the operator [G.sub.[OMEGA]] as a successive application of the truncated direct and truncated inverse Fourier transforms.

Similar to the examples studied in [30, 31, 21, 22], functions from the range of G[OMEGA], Rg([G.sub.[OMEGA]]), are well approximated by the elements of the span of the first few prolates [[psi].sub.l], l = 0, 1,..., L. The number L that provides a sufficient approximation accuracy is often called the number of degrees of freedom and is explicitly expressed in terms of [OMEGA]: L ~ 2[[OMEGA].sup.2]/[pi].

A generalization of the results obtained for prolates to other reproducing kernel spaces is thought to be straightforward [16]. However, even in the case of Hankel-band-limited functions, the number of degrees of freedom and the truncation error bounds have not thus far been computed. The relevant analysis will be presented below. The discussion will primarily focus on the Hankel transform analogue of the operator [G.sub.[OMEGA]] and its range.

2 TFT eigenfunctions

Let us return to prolates and their definition via Eq. (2). The eigenvalues [[mu].sub.l] can be shown to be simple. However except for the first L ~ 2[[OMEGA].sup.2]/[pi] of them, these eigenvalues are numerically indistinguishable from zero, while for the first L eigenvalues |[[mu].sub.l]| [approximately equal to] [27, 16].

For prolates [[psi].sub.l], the Fourier inversion formula takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

If the functions [[psi].sub.l] are normalized by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

they form an orthonormal basis in [L.sub.2]([I.sub.[OMEGA]]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Their analytical continuations then form an orthogonal basis in [PW.sub.[OMEGA]] as well:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [[gamma].sub.l] = [|[[micro.sub.l]]|.sup.2]/2 [pi], l = 0, 1,..., are the eigenvalues of the operator [G.sub.[OMEGA]], so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

One can choose the functions [[psi].sub.l] to be real, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

and therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As shown in [27], [[gamma].sub.0] yields the largest possible value for the ratio [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] among the elements of [PW.sub.[OMEGA]]. In general, we denote by [PW.sup.l.sub.[OMEGA]] the orthogonal complement to Span{[[psi].sub.0], [[psi].sub.1],..., [[psi].sub.l-1]} in [PW.sub.[OMEGA]], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the convolution of [[psi].sub.l] with a sinc-kernel we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

This, in addition to (4)-(5), induces one more orthogonality relation [30, 31]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

along with the equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Integrating the latter equality over [I.sub.[OMEGA]] yields for the trace of [G.sub.[OMEGA]]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

the same value is given by integration of the squared sinc-function, namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, Eq. (8) also gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence the following estimate holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

with some positive constants [B.sub.1] and [B.sub.2].

Combined with (10), the estimate (11) explains (6): each summand [[gamma].sub.l] (1-[[gamma].sub.l]) is small, i.e. [[gamma].sub.l] is close either to zero or to one, besides in view of Eq. (10) the number of eigenvalues [[gamma].sub.l] close to one is restricted to L = 2 [[OMEGA].sup.2]/[pi], for details see [16].

The same L defines the number of the essential samples in the reconstruction formula written in terms of TFT eigenfunctions [30, 31]. On substituting expansion (8) in (1) one obtains for f(x) [member of] [PW.sub.[OMEGA]]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Here, the order of double summation is interchangeable, both double series converging pointwise and in the [L.sub.2]-norm. Because of the double summation, series in the sampling formula (12) look more cumbersome than (1), yet in practical calculations the latter series may be more suitable than the classical one, provided that the function f(x) decays rapidly outside the interval [I.sub.[OMEGA]]. If this is the case, then to a very high accuracy the infinite sums in (12) can be truncated to l [less than or equal to] 2 [[OMEGA].sup.2]/[pi] and |k| [less than or equal to] [[OMEGA].sup.2]/[pi]. As was mentioned above, the relevant proofs for particular classes of band-limited functions can be found in [30, 31, 21, 22].

The new sampling formula (12) applied to [[psi].sub.p] provides one more orthogonality relation [30, 31], namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and as a result

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3 Hankel-band-limited Functions

By a Hankel-band-limited function we understand a function expressed through the Finite (Truncated) Hankel Transform:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where [J.sub.m] is the m-th order Bessel function of the 1st kind , m = 0, 1, 2,... , and g([rho]) = [H.sub.m][h](p) stands for the Hankel transform of h(r), i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] evidently [H.sub.m][h]([rho]) = 0 for [rho] > a. Hankel-band-limited functions serve as radial parts of rotationally symmetric 2D-functions with Fourier transform supported on a domain confined to a circle of radius a,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Eq. (13) is obtained by substituting f(r) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

into Eq. (14). One also takes into account the integral representation of the Bessel function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as well as the identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

that holds for any Hankel-band-limited function [32, 28]. Changing the order of integration in the right-hand side of (15) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

see e.g. [24].

Thus Hankel-band-limited functions form a reproducing kernel Hilbert space with a reproducing kernel [K.sub.am](r,p), for details see, e.g. [36, 29, 10, 11]. As follows from the representation (16), [square root of r h(r) [member of] [C.sup.[infinity]]([R.sub.+])].

4 Bessel functions

Before turning to the sampling theorem in the space [B.sub.am] of Hankel-band-limited functions, we shall address a few more properties of the Bessel functions of the 1st kind necessary for future discussion. Most of them are well-known and have been proved elsewhere (see e.g. I32, 28, 24]), and we shall only mention them briefly; the others, like a modification of the Prufer angle associated with Bessel's equation, will be considered in more detail.

4.1 Modified Prufer angle associated with bounded solutions of Bessel's equation

Hereafter m = 0, 1, 2,..., is fixed. Let us introduce functions [theta](r) and A(r), r > 0, as solutions to the ODEs

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

respectively. By direct substitution, one verifies that the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

satisfies Bessel's equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

and therewith

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Here [PSI](r) := r + [theta](r) is the modified Prufer angle of G(r).

All solutions of Bessel's equation which are finite at r = 0 possess the same Prufer angle, [PSI](r), that in the neighborhood of r = 0 satisfies the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

with [a.sub.0] = m + 1/2, [a.sub.1] = 1 / 2 (m + 1), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The series in the right-hand side of (24) converges absolutely and uniformly on [0, [r.sub.0]], provided that |[a.sub.1]|[r.sub.0.sup.2]/2 < 1, since for any k we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Note that the Prufer angle is defined up to a summand k[pi], k [member of] Z. By integrating Eq. (19) with the initial condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

the Prufer angle associated with bounded solutions of (22) is uniquely defined on [[r.sub.0], [infinity]).

Unlike the Prufer angle, the function A(r) specifies a particular bounded solution of Bessel's equation. Thus the conventional Bessel function of the 1st kind [J.sub.m](r) is specified according to its asymptotic behaviour at infinity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

which means that for the corresponding function A(r)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

cf. Eq. (20).

In what follows the functions [theta](r) and A(r) stand for the solutions of Eqs. (19), (20) subject to the conditions (25), (27); as well as the Prufer angle [PSI](r), they correspond to the function [J.sub.m](r). Evidently A(r) is bounded on the interval [1, [infinity]): [bar.A] [less than or equal to] |A (r)| [less than or equal to] [bar.A] for some [bar.A] and [A.bar], say [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We shall use the representation of [J.sub.m](r) through its Prufer angle and A(r) in order to obtain some useful estimates of integrals containing Bessel functions.

4.2 Lommel integrals

The integrals arising most frequently in the subsequent discussion are of the Lommel type [24]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Obviously, if r < 1/a, [I.sub.L] is less than 1/a, since |[J.sub.k]| [less than or equal to] 1, k = 0, 1,.... In order to obtain an upper bound on [I.sub.L] for r > 1/a, one substitutes the functions [PHI] and A into (28):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

4.3 Zeros of the Bessel functions

Let us denote the zeros of the Bessel function [J.sub.m](r) by [r.sub.mn], n = 0, 1, 2,.... As is well-known, [r.sub.mn] = [pi] (n + m + 1/4) + O(1), r [right arrow] [infinity]. Using Eq. (19) one deduces a sharper asymptotic behaviour of the zeros [r.sub.mn] at infinity. Each time when [J.sub.m](r) = 0, the Prufer angle crosses the line ([PHI]) = [pi] k, k [member of] Z, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If m = 1, 2,..., we have [theta]'([xi]) [less than or equal to] 0, and hence the distance between two neighboring zeros [r.sub.mn] and [r.sub.mn+l] is never less than [pi]. In the case m = 0, we still can write[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As a result we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

Later we will also use the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

5 The Finite Hankel Transform and its eigenfunctions

Let [H.sub.am] be the THT operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

defined on functions g(r) [member of] [L.sup.2]([I.sub.a]), with [I.sub.a] = (0, a). Let also [T.sub.aml](r) = [T.sub.l]/(r) be its eigenfunctions,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

associated with the eigenvalues [Y.sub.aml] = [Y.sub.l] ordered in accordance with their absolute magnitude:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Evidently [H.sub.am] = [H.sup.*.sub.am] and the functions [T.sub.l] can be chosen real. They are at the same time eigenfunctions of the integral operator [K.sub.am] := [H.sub.am] o [H *.sub.am] = [H.sub.am.sup.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The similarity between the properties of TFT and THT eigenfunction was discussed in detail in [26], see also [15, 13].

Being the eigenfunctions of a self-adjoint compact operator, [T.sub.l] form an orthonormal basis in [L.sup.2]([I.sub.a]), provided that they are normalized by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

Further, Eq. (32) not only defines [T.sub.l] inside the interval [I.sub.a], but also its analytic continuation onto the positive semi-axis R+. For the Hankel transform of the function [T.sub.l] continued to [R.sub.+]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

holds, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In addition to the orthogonality on [I.sub.a], Tz are pairwise orthogonal on [R.sub.+]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

In particular, this means that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[parallel]*[parallel].sub.a] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the norms in [L.sup.2] ([I.sub.a]) and in [B.sub.am] respectively.

Similarly to [[psi].sub.l], the eigenfunction [T.sub.l] yields a maximum of the ratio [parallel]h[parallel]a/[parallel][R.sub.+] among all h [member of] [B.sub.m] orthogonal to Span{[T.sub.0], [T.sub.1],...,[T.sub.p-1]} [26].

The completeness of the functions [T.sub.l] in [L.sup.2] ([I.sub.a]) implies that they also form a basis in [B.sub.am]. Indeed, let h [member of] [B.sub.am] and let [h.sub.l] be the Fourier coefficients of its Hankel transform in terms of [T.sub.l] in [L.sup.2]([I.sub.a]), i.e. [h.sub.l] = [<[H.sub.m][h], [T.sub.l] >.sub.a], l = 0, 1,.... Then in view of (15)-(32), and (34), for any L [member of] N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As an immediate consequence of the completeness, the following evident but important expansions hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

Fixing one variable, say r [member of] [R.sub.+], the expansions (36) and (37) are valid in [L.sup.2]([I.sub.a]) and [L.sup.2] ([R.sub.+]), respectively in terms of the remaining variable p. Moreover, the series in the right-hand side of (37) converges uniformly on p [member of] [R.sub.+]. This is readily seen from the obvious relations (below B is the same as in (29)):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

Expansion (36) allows us to compute the trace of [H.sub.am], namely for p = r

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly from (37) we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

In [15] the asymptotic behaviour of the eigenvalues [[??].sub.avl] was studied both for large l and for large a, and it was found that

[[??].sub.avl] [right arrow] 0, as l [right arrow] [infinity] and [[??].sub.avl] [right arrow] [(-1).sup.l] as a [right arrow] [infinity];

here v is not necessarily an integer, v = 1/2 corresponds to the TFT case. The arguments of [15] did not involve any estimates generalizing formulae (10), (11), however such a generalization seems reasonable: similarly to the TFT case, the relevant estimates would allow us to determine the number of eigenvalues [[??].sub.l] essentially different from zero, i.e. to determine the number of degrees of freedom in [B.sub.am].

6 Number of degrees of freedom

Hereafter we assume that a >> 1. The internal integral in (39) is the same Lommel integral as in (28):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

On substituting the asymptotics (26) and the normalization condition [32]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

into (40), we conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

As the next step we formulate and prove an analogue of the inequality (11). To this end, we first show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

This equality follows from Eqs. (39), along with Parseval's theorem which in the case of the Hankel transform claims that [||h||.sub.[infinity]] = [||[H.sub.m][h]||.sub.a] [for all]h [member of] [B.sub.am] [28]. In particular,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies (43). One can rewrite the right-hand side of Eq. (43) differently:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

We use Eq. (44) in the proof of the following proposition which generalizes the inequality (11).

Proposition 1. There exist constants [[bar.B].sub.1] and [[bar.B].sub.2] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

Proof. We first consider the subdomain [0, 1] x [[a.sup.2], [infinity]) partitioned off from the whole integration area in the right hand side of (44). Since for any k, [[J.sub.k](r)l [less than or equal to] 1, and due to (26) there exists a constant [C.sub.1] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting (21) and (23) into the remaining part of the integral in (44) yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To obtain an estimate of the integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we use the definition of the Prufer angle, namely that [PHI]([rho]) - [PHI](r) = ([rho] - r) + ([theta]([rho]) - [theta](r)) and the boundedness of the function ,4 on the interval [1, [infinity]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the first term of the latter integral we apply the same estimate as in (11):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Further, due to Eq. (19) for [rho] > r

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

holds, and hence a constant C4 can be found such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which completes the proof of the inequality (45), since the integrals [I.sub.1], [I.sub.2], [I.sub.3] compose the right hand side of Eq. (44), and as a result, the sum [summation over [infinity] over [l=0]]([[??].sup.2.sub.l] - [[??].sup.4].sub.l]).

Thus we can conclude that the number of the eigenvalues [[??].sub.l] essentially different from zero does not exceed [a.sup.2] / [pi] - m/2 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

The proof repeats that for TFT in [16].

7 Kramer's Sampling Theorem in [B.sub.ma]

For an arbitrary function h [member of] [B.sub.ma], Kramer's sampling formula can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

see [29, 10, 11].

The representation (47) is easy to obtain by expanding the function g = [H.sub.m][h] into the Fourier series in the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (which form an orthonormal basis in [L.sub.2] ([I.sub.a])), and then integrating the series versus the Hankel transform kernel. According to Eq. (13), the Fourier coefficients of the function g are the samples h ([r.sub.mn] / a), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

The above series converges in [L.sub.2] ([I.sub.a]) and by Parseval's theorem, this implies the convergence of the sampling series (47) in [L.sub.2] ([R.sub.+])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as well as the uniform convergence on [R.sub.+] (cf. Section 3),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with the constant B being the same as in (28).

On substituting [rho] = [r.sub.mn]/a in (37) one rewrites the sampling formula (47) in terms of [T.sub.l] (r):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

Similar to the case of prolates, the sampling formula with respect to the THT eigenfunctions provides new discrete orthogonality relations. Let us multiply both sides of the equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by [T.sub.s] (r), and integrate them over [0, a]. This yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

Another discrete orthogonality relation is easily derived from the expansion of the reproducing kernel [K.sub.am]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

8 Truncation numbers

As in the modified WKS formula, the summations over n and l in (49) are interchangeable, i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52)

One derives the latter sampling formula from the Fourier expansion of the function g = [H.sub.m][h] in terms of [T.sub.l], the Fourier coefficients then being

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where inside the interval [I.sub.a] the functions [T.sub.l] are replaced by their expansions (48).

Similarly to (47), the sampling series (49) and (52) converge both in [L.sub.2] ([R.sub.+]) and uniformly on [R.sub.+]. However because of the double summation, they look much more complicated than (47), and without some additional assumptions on the sampled function seem not so worthy. Yet, for the functions from the range of [K.sub.am], the representation (49) becomes more useful than (47). The proposition below shows that the contribution from [T.sub.l] at indices l > L in the Fourier expansion of h [member of] [R.sub.g] ([K.sub.am]) is at most [[??].sub.L+1] as large as that in the associated function G = [[K.sup.-1].sub.am] (h).

Proposition 2. Let h [member of] Rg([K.sub.am]), i.e. h(r) = [[[integral].sup.a].sub.0] [K.sub.am] (r [rho])G([rho]) d [rho] for some G [member of] [L.sub.2] ([I.sub.a]). Denote by [h.sub.1], [G.sub.l] the Fourier coefficients (in terms of [T.sub.l]) of the function h and G, respectively (we remind that for h its Fourier expansion is valid on the [R.sub.+], while for G only on [I.sub.a]). Then for the truncation error caused by neglect of the contribution from [T.sub.l] at l > L

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. For the proof, we notice that inside the interval [I.sub.a] the Hankel transform [H.sub.m] [h] ([eta]) coincides with the band-limited function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and hence is itself representable as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53)

so that [h.sub.l] = [[??].sup.2].sub.l] [G.sub.l]. Therefore on account of Parseval's theorem and the orthogonality relation (33),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Looking back at the relations (46), we see that the reasonable truncation number is L ~ [a.sup.2] / [pi] - m/2.

Now let us compute the error caused by the truncation of the inner sum in (52),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here again the functions from Rg([K.sub.am]) are advantageous over others in [B.sub.am] because of the convergence rate of their samples.

Proposition 3. Let h [member of] Rg([K.sub.am]), then samples h ([r.sub.mn]/a) decay at infinity as 1/[r.sub.mn].

Proof. To show that, we write for [a.sup.2] < [r.sub.mN] < [r.sub.mn]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (54)

Note that [T.sub.l] are also in Rg([K.sub.am]), and hence the above estimate holds for them with [[??].sup.-2].sub.l] replacing [[parallel]G[parallel].sub.a].

As follows from (30), [r.sub.mn] = [r.sub.mN] + [pi] (n - N) + O (1/[n.sup.2]), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (55)

For the truncation error [[epsilon].sub.LN] one writes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56)

Summing up the relations (54), (55) and (56), we obtain an upper bound on the error [[epsilon].sub.LN],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

Here it is worth noticing that, if a is large and [r.sub.mN] > [a.sup.2], the contribution from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

into the truncation error [[epsilon].sub.LN] is essential only for l ~ [a.sup.2]/[pi] - m/2 and the following proposition holds:

Proposition 4. Let [r.sub.mN] > [a.sup.2]. There exists a constant 19 such that for any l = 0,1,...,

[[SIGMA].sub.l] [is less than or equal to] D [[??].sub.l] [square root of 1- [[[??].sup.2].sub.l].

Proof. To begin with, any THT eigenfunction satisfies the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58)

The proof of (58) is almost identical to that of Lemma 4 in [30], and is therefore omitted here.

If m [greater than or equal to] I, we have in the right-hand side of (58)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here we have used Eq. (41), Bessel's equation written in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the relation [[integral].sup.[infinity].sub.0] r [J.sub.m] ([r.sub.[rho]]) [J.sub.m] ([r.sub.[xi]]) dr = [delta]([rho] - [xi])/[xi].

Even if m = 0, one can still obtain a similar estimate, e.g. by proving that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with a constant [D.sub.1] independent of l.

Let [[DELTA].sub.m] be the smallest distance between two zeros of [J.sub.m] (r), then it follows from (58)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In view of (31) we write (cf. [30]) for m [greater than or equal to] 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (59)

The first term in the right-hand side of (59) is dominated by the second one, whence the result. For m = 0 the parameter a is to be replaced with [square root of ([D.sub.1])]. []

In any case [[??].sub.l] [square root of (1 - [y.sup.2.sub.l])] is very small for large a and I essentially different

from [a.sup.2]/[pi] - m/2. Thus the optimal truncation number N (i.e. the number of employed samples of the reconstructed function h [member of] Rg([K.sub.am])) is the smallest one for which [r.sub.mN] > [a.sup.2], that is N ~ [a.sup.2]/[pi] - m.

9 Important remarks

Our first important remark concerns Hankel-band-limited functions that are not in the range of [K.sub.am]. If such a function can still be well approximated by a linear combination of the first L ~[a.sup.2]/[pi] - m/2 THT eigenfunctions [T.sub.l], then again the main contribution to the truncation error ELN comes from [T.sub.l] at l ~ L, and although the samples h ([r.sub.mn]/a) may decay not as fast as previously, one can still write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Not every Hankel-band-limited function is well approximated by a truncated sampling sum (49). As an obvious counter example one can consider here the kernel [K.sub.am] ([??], r). For any fixed [??] > 0 this function is evidently Hankel-band limited, however, as is readily seen from (37),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here the error [epsilon]L converges to zero as L increases, but not as fast as [[??].sub.L].

Additionally we remark that by (46) we have obtained an explicit bound M ~ 2 [a.sup.2]/[pi] on the number of effective angular momenta m required in sampling of two-dimensional band-limited signals.

Finally we would like to emphasize that the range of operator [K.sub.am] plays in [B.sub.am] the role similar to that of Rg([G.sub.[OMEGA]]) in the Paley-Wiener space. With minor changes the results obtained for Rg([K.sub.am]) can be transferred to Rg([G.sup.[OMEGA]]) and the functions from Rg([G.sub.[OMEGA]]) to a high accuracy can be reconstructed from their 2[[OMEGA].sup.2]/Tr samples through the first 2[[OMEGA].sup.2]/[pi] TFT eigenfunctions.

10 Evaluation of the THT eigenfunctions

As proved in [26], [T.sub.l (r) are the bounded solutions of the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e., the eigenfunctions of a singular self-adjoint Sturm-Liouville problem. The author of [26] called these solutions generalized prolate spheroidal functions, and investigated various asymptotic cases, e.g. for both small and large radii a (see also [15, 13]). Based on the discussion in [26], as well as on the approach developed in [1, 2, 20] for numerical solution of singular self-adjoint Sturm-Liouville problems, a new efficient, robust and accurate numerical technique was recently invented for evaluation of THT eigenfunctions and eigenvalues. In [19] its tentative version was briefly discussed and first preliminary calculations illustrating the facilities of the method were presented. In [4] various numerical methods for solving the problem (60) are discussed in detail and compared. The advantages are that the eigenvalues of problem (60) are well-resolved and that the original integral equation (32) is not used for calculations but rather for their verification. The numerical approaches presented in [4] are accurate, robust and efficient and allow calculation of the THT eigenfunctions for a wide range of parameters.

ACKNOWLEDGEMENT

The author is grateful to the editor, Prof. Achmed Zayed, and the anonymous referee for their valuable comments that have substantively improved the manuscript.

References

[1] A. A. Abramov, A. L. Dyshko, N. B. Konyukhova, T. V. Pak, and B. S. Pariiskii, Evaluation of prolate spheroidal function by solving the corresponding differential equations, U.S.S.R. Comput. Math. and Math. Phys., 24(i), 1-il, 1984.

[2] A. A. Abramov, A. L. Dyshko, N. B. Konyukhova, and T. V. Levitina, Computation of radial wave functions for spheroids and triaxial ellipsoids by the modified phase function method, Comput. Math. and Math. Phys., 31(2), 25-42, 1991.

[3] C. Aime, Apodized apertures for solar coronagraphy, Astronomy and Astrophysics, 467(1), 317-325, 2007.

[4] P. Amodio, T. Levitina, G. Settanni, and E. B. Weinmfiller, On the calculation of the finite Hankel transform eigenfunctions (submitted to Y. Appl. Math. & Computing).

[5] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68, 337-404, 1950.

[6] V. N. Beskrovny and M. I. Kolobov Quantum-statistical analysis of superresolution for optical systems with circular symmetry, Phys. Rev., 78, 043824(1-11), 2008.

[7] W. N. Everitt and G. Nasri-Roudsari, Interpolation and sampling theories, and linear ordinary boundary value problems. Ch. 5 in J. R. Higgins and R. L. Stens, editors, Sampling Theory in Fourier and Signal Analysis: Advanced Topics, Oxford University Press, Oxford, 1999.

[8] W. N. Everitt, G. Nasri-Roudsari, and J. Rehberg, A note on the analytic form of the Kramer sampling theorem, Results Math., 34(3-4), 310-319, 1998.

[9] J.C. Heurtley, Hyperspheroidal functions - Optical resonators with circular mirrors. In Proc. Symposium on Quasi-Optics, Polytechnic Press, New York, 367-375, 1964.

[10] J. R. Higgins, An interpolation series associated with the Bessel-Hankel transform, Journal of the London Mathematical Society, 5, 707-714, 1972.

[11] J. R. Higgins, Five short stories about the cardinal series, Bull. Amer. Math. Soc. (N.S.), 12(1), 45-89, 1985.

[12] A. J. Jerri, The. Shannon sampling theorem-Its various extensions and applications: A tutorial review, Proc. IEEE, 65(11), 1565-1596, 1977.

[13] I. V. Komarov, L. I. Ponomarev and S. Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions, [in Russian], Nauka, Moscow, 1976.

[14] H. P. Kramer, A generalized sampling theorem, Your. Math. Phys, 38, 68-72, 1959.

[15] N. V. Kuznetsov, Eigenfunctions of a certain integral equation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov, 17, 66-150, 1970.

[16] H. J. Landau, Sampling, data transmission, and the Nyquist rate, Proc. IEEE, 55, 1701-1706, 1967.

[17] H. Landau and H. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty II, Bell Sys. Tech. Y., 40, 65-84, 1961.

[18] H. Landau and H. Pollak, Prolate spheroidal wavefunctions, Fourier analysis and uncertainty III, Bell Sys. Tech. Y., 41, 1295-1336, 1962.

[19] B. Larsson, T. V. Levitina and E. J. Brandas, Eigenfunctions of the 2D finite Fourier transform, Y. Comp. Meth. Sci. & Engrg., 4, 135-148, 2004.

[20] T. V. Levitina and E J. Brandas , Computational techniques for prolate spheroidal wave functions in signal processing, J. Comp. Meth. Sci. & Engrg., 1,287-313, 2001.

[21] T. V. Levitina and E. J. Brandas, Sampling formula for convolution with a prolate, International Journal of Computer Mathematics, 85, 487-496, 2008.

[22] T. V. Levitina and E. J. Brandas, Filter diagonalization: Filtering and postprocessing with prolates, Computer Physics Communications, 180(9), 1448-1457, 2009.

[23] R. T. W. Martin, Symmetric operators and reproducing kernel Hilbert spaces, Complex Anal. Oper. Theory, 4, 845-880, 2010.

[24] N. W. McLachlan, Bessel Functions for Engineers, Clarendon Press, Oxford, 1934.

[25] S. S. Sherif, M. R. Foreman and P. Torok, Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system, Optics Express, 16(5), 3397-3407, 2008.

[26] D. Slepian, Prolate spheroidal wavefunctions, Fourier analysis and uncertainty, IV: Extensions to many dimensions; generalized prolate spheroidal functions, Bell Sys. Tech. J., 43, 3009-3058, 1964.

[27] D. Slepian and H. O. Pollak, Prolate spheroidal wavefunctions, Fourier analysis and uncertainty, I, Bell Syst. Tech. J., 40(1), 43-64, 1961.

[28] I. Sneddon, Fourier transforms, McGraw-Hill, New York, 1951.

[29] H. Stark, Sampling theorems in polar coordinates, J. Opt. Soc. Amer., 69(11), 1519-1525, 1979.

[30] G. G. Walter and X. Shen, Sampling with prolate spheroidal wave functions, Sampl. Theory Signal Image Process., 2(1), 25-52, 2003.

[31] G. G. Walter, X. Shen, Wavelets based on prolate spheroidal wave functions, J. Fourier Anal. Appl., 10(1), 1-26, 2004.

[32] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1962.

[33] L. A. Weinstein, Open Resonators and Open Waveguides, Golem Press, Boulder, Colorado, 1969.

[34] H. Xiao, V. Rokhlin and N. Yarvin, Prolate spheroidal wave functions, quadrature and interpolation, Inverse Problems, 17, 805-838, 2001.

[35] A.I. Zayed, On Kramer's sampling theorem associated with general Strum-Liouville problems and Lagrange interpolation, SIAM Journal on Applied Mathematics, 51(2), 575-604, 1991.

[36] A. I. Zayed, A generalization of the prolate spheroidal wave functions, Proc. Amer. Math. Soc., 135, 2193-2203, 2007.

(1) In order to avoid ambiguity, the abbreviation TFT (THT) stands here for the Finite Fourier (Hankel) Transform, as the acronym FFT is usually used for the Fast Fourier Transform.

Tatiana Levitina

Institut Computational Mathematics, Technische Universitat Braunschweig

Braunschweig, D-38106, Germany

t.levitina@tu-bs.de

In this paper, we study the properties of the eigenfunctions of the finite Hankel transform. We deduce a sampling series in terms of these functions for Hankel-band-limited signals and derive bounds for the truncation error of the sampling series.

Key words and phrases : Reproducing kernel space, Kramer sampling theorem, finite Hankel transform.

2000 AMS Mathematics Subject Classification--41A05, 65D05, 65D20.

1 Introduction

The classical Whittaker-Kotel'nikov-Shannon (WKS) sampling theorem says that any function f from the Paley-Wiener space [PW.sub.[OMEGA]], i.e. any f [member of] [L.sup.2](-[infinity], [infinity]) representable through its Fourier transform F[f] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

can be reconstructed from its equally spaced samples,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The series in the right-hand side converges both in the [L.sub.2]-norm and uniformly on (-[infinity], [infinity]). This follows from the fact that F[f] [member of] [L.sup.2]([OMEGA], [OMEGA]) and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Another sampling formula for f(t), which was invented and studied in [30, 31], is based on the eigenfunctions [[psi].sub.l]([OMEGA], x) = [[psi].sub.l] (x) of the Finite (Truncated)

Fourier Transform [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The eigenfunctions

[[psi].sub.l](x) are first defined inside the interval [I.sub.[OMEGA]] = [-[OMEGA], [OMEGA]] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

and are then continued analytically to the entire real axis according to (2). The associated TFT eigenvalues [[mu].sub.l] = [[mu].sub.l]([OMEGA]) are ordered by ma[G.sub.[OMEGA]]itude, [square root 2 [pi]] |[[mu].sub.0]| > |[[mu].sub.1]| > ... > 0.

Due to their unusual and interesting features, the TFT eigenfunctions, also known as 'prolates', have long attracted attention; we refer the reader to the classical works by Slepian, Pollak, and Landau [27, 16, 17, 18], as well as the papers [30, 31, 34] and the literature cited therein. This list of references is far from complete. The properties of prolates [[psi].sub.l] are widely discussed in the literature. Nevertheless we shall proceed with a short survey of those properties which are related to the fact that prolates form an orthonormal basis in a reproducing kernel space [5]. In such spaces Kramer's generalization of the WKS sampling theorem is valid. Kramer's theorem expands the variety of functions that can be recovered from their sampled values by means of orthogonal sampling formulae, and also establishes general conditions allowing such a recovery (see e.g. [14, 12, 35, 8, 7, 23]).

The essence of the general theory is summarized in a recent paper [36], where an algorithm to generate systems of functions possessing the above--mentioned properties of prolates is provided. We shall illustrate the general description of [36] with the example of the Finite (Truncated) Hankel Transform (THT) and its eigenfunctions. These functions are extensively used now in various areas that range from applied physics to computer science. The number of publications in which they are employed is so large that any comprehensive survey of modern applications would be a very complicated task going far beyond the present study. Here we only note that historically the THT eigenfunctions were introduced almost simultaneously in three pioneering works [26, 9, 33], each being a source of many subsequent publications on numerical optical analysis, such as both forward and inverse analysis of high numerical aperture focusing systems, calculation of mode patterns and losses in optical resonators with spherical mirrors. Among the most recent papers see, e. g. [25, 6, 3].

In the present work we pay special attention to the bounds of the error induced by the sampling series truncation. In [30, 31, 211 22] the truncation error estimates were obtained for particular classes of band--limited functions and some illustrative examples were given. In what follows we shall be concerned with another class of band-limited functions, namely with the range of the operator [G.sub.[OMEGA]] := (1/2 [pi])[F.sup.*.sub.[OMEGA]] o [F.sub.[OMEGA]], i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

One can also interpret the operator [G.sub.[OMEGA]] as a successive application of the truncated direct and truncated inverse Fourier transforms.

Similar to the examples studied in [30, 31, 21, 22], functions from the range of G[OMEGA], Rg([G.sub.[OMEGA]]), are well approximated by the elements of the span of the first few prolates [[psi].sub.l], l = 0, 1,..., L. The number L that provides a sufficient approximation accuracy is often called the number of degrees of freedom and is explicitly expressed in terms of [OMEGA]: L ~ 2[[OMEGA].sup.2]/[pi].

A generalization of the results obtained for prolates to other reproducing kernel spaces is thought to be straightforward [16]. However, even in the case of Hankel-band-limited functions, the number of degrees of freedom and the truncation error bounds have not thus far been computed. The relevant analysis will be presented below. The discussion will primarily focus on the Hankel transform analogue of the operator [G.sub.[OMEGA]] and its range.

2 TFT eigenfunctions

Let us return to prolates and their definition via Eq. (2). The eigenvalues [[mu].sub.l] can be shown to be simple. However except for the first L ~ 2[[OMEGA].sup.2]/[pi] of them, these eigenvalues are numerically indistinguishable from zero, while for the first L eigenvalues |[[mu].sub.l]| [approximately equal to] [27, 16].

For prolates [[psi].sub.l], the Fourier inversion formula takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

If the functions [[psi].sub.l] are normalized by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

they form an orthonormal basis in [L.sub.2]([I.sub.[OMEGA]]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Their analytical continuations then form an orthogonal basis in [PW.sub.[OMEGA]] as well:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [[gamma].sub.l] = [|[[micro.sub.l]]|.sup.2]/2 [pi], l = 0, 1,..., are the eigenvalues of the operator [G.sub.[OMEGA]], so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

One can choose the functions [[psi].sub.l] to be real, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

and therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As shown in [27], [[gamma].sub.0] yields the largest possible value for the ratio [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] among the elements of [PW.sub.[OMEGA]]. In general, we denote by [PW.sup.l.sub.[OMEGA]] the orthogonal complement to Span{[[psi].sub.0], [[psi].sub.1],..., [[psi].sub.l-1]} in [PW.sub.[OMEGA]], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the convolution of [[psi].sub.l] with a sinc-kernel we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

This, in addition to (4)-(5), induces one more orthogonality relation [30, 31]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

along with the equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Integrating the latter equality over [I.sub.[OMEGA]] yields for the trace of [G.sub.[OMEGA]]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

the same value is given by integration of the squared sinc-function, namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, Eq. (8) also gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence the following estimate holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

with some positive constants [B.sub.1] and [B.sub.2].

Combined with (10), the estimate (11) explains (6): each summand [[gamma].sub.l] (1-[[gamma].sub.l]) is small, i.e. [[gamma].sub.l] is close either to zero or to one, besides in view of Eq. (10) the number of eigenvalues [[gamma].sub.l] close to one is restricted to L = 2 [[OMEGA].sup.2]/[pi], for details see [16].

The same L defines the number of the essential samples in the reconstruction formula written in terms of TFT eigenfunctions [30, 31]. On substituting expansion (8) in (1) one obtains for f(x) [member of] [PW.sub.[OMEGA]]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Here, the order of double summation is interchangeable, both double series converging pointwise and in the [L.sub.2]-norm. Because of the double summation, series in the sampling formula (12) look more cumbersome than (1), yet in practical calculations the latter series may be more suitable than the classical one, provided that the function f(x) decays rapidly outside the interval [I.sub.[OMEGA]]. If this is the case, then to a very high accuracy the infinite sums in (12) can be truncated to l [less than or equal to] 2 [[OMEGA].sup.2]/[pi] and |k| [less than or equal to] [[OMEGA].sup.2]/[pi]. As was mentioned above, the relevant proofs for particular classes of band-limited functions can be found in [30, 31, 21, 22].

The new sampling formula (12) applied to [[psi].sub.p] provides one more orthogonality relation [30, 31], namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and as a result

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3 Hankel-band-limited Functions

By a Hankel-band-limited function we understand a function expressed through the Finite (Truncated) Hankel Transform:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where [J.sub.m] is the m-th order Bessel function of the 1st kind , m = 0, 1, 2,... , and g([rho]) = [H.sub.m][h](p) stands for the Hankel transform of h(r), i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] evidently [H.sub.m][h]([rho]) = 0 for [rho] > a. Hankel-band-limited functions serve as radial parts of rotationally symmetric 2D-functions with Fourier transform supported on a domain confined to a circle of radius a,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Eq. (13) is obtained by substituting f(r) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

into Eq. (14). One also takes into account the integral representation of the Bessel function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as well as the identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

that holds for any Hankel-band-limited function [32, 28]. Changing the order of integration in the right-hand side of (15) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

see e.g. [24].

Thus Hankel-band-limited functions form a reproducing kernel Hilbert space with a reproducing kernel [K.sub.am](r,p), for details see, e.g. [36, 29, 10, 11]. As follows from the representation (16), [square root of r h(r) [member of] [C.sup.[infinity]]([R.sub.+])].

4 Bessel functions

Before turning to the sampling theorem in the space [B.sub.am] of Hankel-band-limited functions, we shall address a few more properties of the Bessel functions of the 1st kind necessary for future discussion. Most of them are well-known and have been proved elsewhere (see e.g. I32, 28, 24]), and we shall only mention them briefly; the others, like a modification of the Prufer angle associated with Bessel's equation, will be considered in more detail.

4.1 Modified Prufer angle associated with bounded solutions of Bessel's equation

Hereafter m = 0, 1, 2,..., is fixed. Let us introduce functions [theta](r) and A(r), r > 0, as solutions to the ODEs

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

respectively. By direct substitution, one verifies that the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

satisfies Bessel's equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

and therewith

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Here [PSI](r) := r + [theta](r) is the modified Prufer angle of G(r).

All solutions of Bessel's equation which are finite at r = 0 possess the same Prufer angle, [PSI](r), that in the neighborhood of r = 0 satisfies the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

with [a.sub.0] = m + 1/2, [a.sub.1] = 1 / 2 (m + 1), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The series in the right-hand side of (24) converges absolutely and uniformly on [0, [r.sub.0]], provided that |[a.sub.1]|[r.sub.0.sup.2]/2 < 1, since for any k we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Note that the Prufer angle is defined up to a summand k[pi], k [member of] Z. By integrating Eq. (19) with the initial condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

the Prufer angle associated with bounded solutions of (22) is uniquely defined on [[r.sub.0], [infinity]).

Unlike the Prufer angle, the function A(r) specifies a particular bounded solution of Bessel's equation. Thus the conventional Bessel function of the 1st kind [J.sub.m](r) is specified according to its asymptotic behaviour at infinity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

which means that for the corresponding function A(r)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

cf. Eq. (20).

In what follows the functions [theta](r) and A(r) stand for the solutions of Eqs. (19), (20) subject to the conditions (25), (27); as well as the Prufer angle [PSI](r), they correspond to the function [J.sub.m](r). Evidently A(r) is bounded on the interval [1, [infinity]): [bar.A] [less than or equal to] |A (r)| [less than or equal to] [bar.A] for some [bar.A] and [A.bar], say [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We shall use the representation of [J.sub.m](r) through its Prufer angle and A(r) in order to obtain some useful estimates of integrals containing Bessel functions.

4.2 Lommel integrals

The integrals arising most frequently in the subsequent discussion are of the Lommel type [24]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Obviously, if r < 1/a, [I.sub.L] is less than 1/a, since |[J.sub.k]| [less than or equal to] 1, k = 0, 1,.... In order to obtain an upper bound on [I.sub.L] for r > 1/a, one substitutes the functions [PHI] and A into (28):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

4.3 Zeros of the Bessel functions

Let us denote the zeros of the Bessel function [J.sub.m](r) by [r.sub.mn], n = 0, 1, 2,.... As is well-known, [r.sub.mn] = [pi] (n + m + 1/4) + O(1), r [right arrow] [infinity]. Using Eq. (19) one deduces a sharper asymptotic behaviour of the zeros [r.sub.mn] at infinity. Each time when [J.sub.m](r) = 0, the Prufer angle crosses the line ([PHI]) = [pi] k, k [member of] Z, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If m = 1, 2,..., we have [theta]'([xi]) [less than or equal to] 0, and hence the distance between two neighboring zeros [r.sub.mn] and [r.sub.mn+l] is never less than [pi]. In the case m = 0, we still can write[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As a result we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

Later we will also use the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

5 The Finite Hankel Transform and its eigenfunctions

Let [H.sub.am] be the THT operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

defined on functions g(r) [member of] [L.sup.2]([I.sub.a]), with [I.sub.a] = (0, a). Let also [T.sub.aml](r) = [T.sub.l]/(r) be its eigenfunctions,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

associated with the eigenvalues [Y.sub.aml] = [Y.sub.l] ordered in accordance with their absolute magnitude:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Evidently [H.sub.am] = [H.sup.*.sub.am] and the functions [T.sub.l] can be chosen real. They are at the same time eigenfunctions of the integral operator [K.sub.am] := [H.sub.am] o [H *.sub.am] = [H.sub.am.sup.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The similarity between the properties of TFT and THT eigenfunction was discussed in detail in [26], see also [15, 13].

Being the eigenfunctions of a self-adjoint compact operator, [T.sub.l] form an orthonormal basis in [L.sup.2]([I.sub.a]), provided that they are normalized by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

Further, Eq. (32) not only defines [T.sub.l] inside the interval [I.sub.a], but also its analytic continuation onto the positive semi-axis R+. For the Hankel transform of the function [T.sub.l] continued to [R.sub.+]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

holds, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In addition to the orthogonality on [I.sub.a], Tz are pairwise orthogonal on [R.sub.+]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

In particular, this means that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[parallel]*[parallel].sub.a] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the norms in [L.sup.2] ([I.sub.a]) and in [B.sub.am] respectively.

Similarly to [[psi].sub.l], the eigenfunction [T.sub.l] yields a maximum of the ratio [parallel]h[parallel]a/[parallel][R.sub.+] among all h [member of] [B.sub.m] orthogonal to Span{[T.sub.0], [T.sub.1],...,[T.sub.p-1]} [26].

The completeness of the functions [T.sub.l] in [L.sup.2] ([I.sub.a]) implies that they also form a basis in [B.sub.am]. Indeed, let h [member of] [B.sub.am] and let [h.sub.l] be the Fourier coefficients of its Hankel transform in terms of [T.sub.l] in [L.sup.2]([I.sub.a]), i.e. [h.sub.l] = [<[H.sub.m][h], [T.sub.l] >.sub.a], l = 0, 1,.... Then in view of (15)-(32), and (34), for any L [member of] N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As an immediate consequence of the completeness, the following evident but important expansions hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

Fixing one variable, say r [member of] [R.sub.+], the expansions (36) and (37) are valid in [L.sup.2]([I.sub.a]) and [L.sup.2] ([R.sub.+]), respectively in terms of the remaining variable p. Moreover, the series in the right-hand side of (37) converges uniformly on p [member of] [R.sub.+]. This is readily seen from the obvious relations (below B is the same as in (29)):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

Expansion (36) allows us to compute the trace of [H.sub.am], namely for p = r

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly from (37) we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

In [15] the asymptotic behaviour of the eigenvalues [[??].sub.avl] was studied both for large l and for large a, and it was found that

[[??].sub.avl] [right arrow] 0, as l [right arrow] [infinity] and [[??].sub.avl] [right arrow] [(-1).sup.l] as a [right arrow] [infinity];

here v is not necessarily an integer, v = 1/2 corresponds to the TFT case. The arguments of [15] did not involve any estimates generalizing formulae (10), (11), however such a generalization seems reasonable: similarly to the TFT case, the relevant estimates would allow us to determine the number of eigenvalues [[??].sub.l] essentially different from zero, i.e. to determine the number of degrees of freedom in [B.sub.am].

6 Number of degrees of freedom

Hereafter we assume that a >> 1. The internal integral in (39) is the same Lommel integral as in (28):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

On substituting the asymptotics (26) and the normalization condition [32]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

into (40), we conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

As the next step we formulate and prove an analogue of the inequality (11). To this end, we first show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

This equality follows from Eqs. (39), along with Parseval's theorem which in the case of the Hankel transform claims that [||h||.sub.[infinity]] = [||[H.sub.m][h]||.sub.a] [for all]h [member of] [B.sub.am] [28]. In particular,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies (43). One can rewrite the right-hand side of Eq. (43) differently:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

We use Eq. (44) in the proof of the following proposition which generalizes the inequality (11).

Proposition 1. There exist constants [[bar.B].sub.1] and [[bar.B].sub.2] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

Proof. We first consider the subdomain [0, 1] x [[a.sup.2], [infinity]) partitioned off from the whole integration area in the right hand side of (44). Since for any k, [[J.sub.k](r)l [less than or equal to] 1, and due to (26) there exists a constant [C.sub.1] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting (21) and (23) into the remaining part of the integral in (44) yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To obtain an estimate of the integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we use the definition of the Prufer angle, namely that [PHI]([rho]) - [PHI](r) = ([rho] - r) + ([theta]([rho]) - [theta](r)) and the boundedness of the function ,4 on the interval [1, [infinity]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the first term of the latter integral we apply the same estimate as in (11):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Further, due to Eq. (19) for [rho] > r

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

holds, and hence a constant C4 can be found such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which completes the proof of the inequality (45), since the integrals [I.sub.1], [I.sub.2], [I.sub.3] compose the right hand side of Eq. (44), and as a result, the sum [summation over [infinity] over [l=0]]([[??].sup.2.sub.l] - [[??].sup.4].sub.l]).

Thus we can conclude that the number of the eigenvalues [[??].sub.l] essentially different from zero does not exceed [a.sup.2] / [pi] - m/2 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

The proof repeats that for TFT in [16].

7 Kramer's Sampling Theorem in [B.sub.ma]

For an arbitrary function h [member of] [B.sub.ma], Kramer's sampling formula can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

see [29, 10, 11].

The representation (47) is easy to obtain by expanding the function g = [H.sub.m][h] into the Fourier series in the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (which form an orthonormal basis in [L.sub.2] ([I.sub.a])), and then integrating the series versus the Hankel transform kernel. According to Eq. (13), the Fourier coefficients of the function g are the samples h ([r.sub.mn] / a), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

The above series converges in [L.sub.2] ([I.sub.a]) and by Parseval's theorem, this implies the convergence of the sampling series (47) in [L.sub.2] ([R.sub.+])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as well as the uniform convergence on [R.sub.+] (cf. Section 3),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with the constant B being the same as in (28).

On substituting [rho] = [r.sub.mn]/a in (37) one rewrites the sampling formula (47) in terms of [T.sub.l] (r):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

Similar to the case of prolates, the sampling formula with respect to the THT eigenfunctions provides new discrete orthogonality relations. Let us multiply both sides of the equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by [T.sub.s] (r), and integrate them over [0, a]. This yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

Another discrete orthogonality relation is easily derived from the expansion of the reproducing kernel [K.sub.am]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

8 Truncation numbers

As in the modified WKS formula, the summations over n and l in (49) are interchangeable, i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52)

One derives the latter sampling formula from the Fourier expansion of the function g = [H.sub.m][h] in terms of [T.sub.l], the Fourier coefficients then being

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where inside the interval [I.sub.a] the functions [T.sub.l] are replaced by their expansions (48).

Similarly to (47), the sampling series (49) and (52) converge both in [L.sub.2] ([R.sub.+]) and uniformly on [R.sub.+]. However because of the double summation, they look much more complicated than (47), and without some additional assumptions on the sampled function seem not so worthy. Yet, for the functions from the range of [K.sub.am], the representation (49) becomes more useful than (47). The proposition below shows that the contribution from [T.sub.l] at indices l > L in the Fourier expansion of h [member of] [R.sub.g] ([K.sub.am]) is at most [[??].sub.L+1] as large as that in the associated function G = [[K.sup.-1].sub.am] (h).

Proposition 2. Let h [member of] Rg([K.sub.am]), i.e. h(r) = [[[integral].sup.a].sub.0] [K.sub.am] (r [rho])G([rho]) d [rho] for some G [member of] [L.sub.2] ([I.sub.a]). Denote by [h.sub.1], [G.sub.l] the Fourier coefficients (in terms of [T.sub.l]) of the function h and G, respectively (we remind that for h its Fourier expansion is valid on the [R.sub.+], while for G only on [I.sub.a]). Then for the truncation error caused by neglect of the contribution from [T.sub.l] at l > L

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. For the proof, we notice that inside the interval [I.sub.a] the Hankel transform [H.sub.m] [h] ([eta]) coincides with the band-limited function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and hence is itself representable as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53)

so that [h.sub.l] = [[??].sup.2].sub.l] [G.sub.l]. Therefore on account of Parseval's theorem and the orthogonality relation (33),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Looking back at the relations (46), we see that the reasonable truncation number is L ~ [a.sup.2] / [pi] - m/2.

Now let us compute the error caused by the truncation of the inner sum in (52),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here again the functions from Rg([K.sub.am]) are advantageous over others in [B.sub.am] because of the convergence rate of their samples.

Proposition 3. Let h [member of] Rg([K.sub.am]), then samples h ([r.sub.mn]/a) decay at infinity as 1/[r.sub.mn].

Proof. To show that, we write for [a.sup.2] < [r.sub.mN] < [r.sub.mn]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (54)

Note that [T.sub.l] are also in Rg([K.sub.am]), and hence the above estimate holds for them with [[??].sup.-2].sub.l] replacing [[parallel]G[parallel].sub.a].

As follows from (30), [r.sub.mn] = [r.sub.mN] + [pi] (n - N) + O (1/[n.sup.2]), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (55)

For the truncation error [[epsilon].sub.LN] one writes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56)

Summing up the relations (54), (55) and (56), we obtain an upper bound on the error [[epsilon].sub.LN],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

Here it is worth noticing that, if a is large and [r.sub.mN] > [a.sup.2], the contribution from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

into the truncation error [[epsilon].sub.LN] is essential only for l ~ [a.sup.2]/[pi] - m/2 and the following proposition holds:

Proposition 4. Let [r.sub.mN] > [a.sup.2]. There exists a constant 19 such that for any l = 0,1,...,

[[SIGMA].sub.l] [is less than or equal to] D [[??].sub.l] [square root of 1- [[[??].sup.2].sub.l].

Proof. To begin with, any THT eigenfunction satisfies the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58)

The proof of (58) is almost identical to that of Lemma 4 in [30], and is therefore omitted here.

If m [greater than or equal to] I, we have in the right-hand side of (58)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here we have used Eq. (41), Bessel's equation written in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the relation [[integral].sup.[infinity].sub.0] r [J.sub.m] ([r.sub.[rho]]) [J.sub.m] ([r.sub.[xi]]) dr = [delta]([rho] - [xi])/[xi].

Even if m = 0, one can still obtain a similar estimate, e.g. by proving that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with a constant [D.sub.1] independent of l.

Let [[DELTA].sub.m] be the smallest distance between two zeros of [J.sub.m] (r), then it follows from (58)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In view of (31) we write (cf. [30]) for m [greater than or equal to] 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (59)

The first term in the right-hand side of (59) is dominated by the second one, whence the result. For m = 0 the parameter a is to be replaced with [square root of ([D.sub.1])]. []

In any case [[??].sub.l] [square root of (1 - [y.sup.2.sub.l])] is very small for large a and I essentially different

from [a.sup.2]/[pi] - m/2. Thus the optimal truncation number N (i.e. the number of employed samples of the reconstructed function h [member of] Rg([K.sub.am])) is the smallest one for which [r.sub.mN] > [a.sup.2], that is N ~ [a.sup.2]/[pi] - m.

9 Important remarks

Our first important remark concerns Hankel-band-limited functions that are not in the range of [K.sub.am]. If such a function can still be well approximated by a linear combination of the first L ~[a.sup.2]/[pi] - m/2 THT eigenfunctions [T.sub.l], then again the main contribution to the truncation error ELN comes from [T.sub.l] at l ~ L, and although the samples h ([r.sub.mn]/a) may decay not as fast as previously, one can still write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Not every Hankel-band-limited function is well approximated by a truncated sampling sum (49). As an obvious counter example one can consider here the kernel [K.sub.am] ([??], r). For any fixed [??] > 0 this function is evidently Hankel-band limited, however, as is readily seen from (37),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here the error [epsilon]L converges to zero as L increases, but not as fast as [[??].sub.L].

Additionally we remark that by (46) we have obtained an explicit bound M ~ 2 [a.sup.2]/[pi] on the number of effective angular momenta m required in sampling of two-dimensional band-limited signals.

Finally we would like to emphasize that the range of operator [K.sub.am] plays in [B.sub.am] the role similar to that of Rg([G.sub.[OMEGA]]) in the Paley-Wiener space. With minor changes the results obtained for Rg([K.sub.am]) can be transferred to Rg([G.sup.[OMEGA]]) and the functions from Rg([G.sub.[OMEGA]]) to a high accuracy can be reconstructed from their 2[[OMEGA].sup.2]/Tr samples through the first 2[[OMEGA].sup.2]/[pi] TFT eigenfunctions.

10 Evaluation of the THT eigenfunctions

As proved in [26], [T.sub.l (r) are the bounded solutions of the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e., the eigenfunctions of a singular self-adjoint Sturm-Liouville problem. The author of [26] called these solutions generalized prolate spheroidal functions, and investigated various asymptotic cases, e.g. for both small and large radii a (see also [15, 13]). Based on the discussion in [26], as well as on the approach developed in [1, 2, 20] for numerical solution of singular self-adjoint Sturm-Liouville problems, a new efficient, robust and accurate numerical technique was recently invented for evaluation of THT eigenfunctions and eigenvalues. In [19] its tentative version was briefly discussed and first preliminary calculations illustrating the facilities of the method were presented. In [4] various numerical methods for solving the problem (60) are discussed in detail and compared. The advantages are that the eigenvalues of problem (60) are well-resolved and that the original integral equation (32) is not used for calculations but rather for their verification. The numerical approaches presented in [4] are accurate, robust and efficient and allow calculation of the THT eigenfunctions for a wide range of parameters.

ACKNOWLEDGEMENT

The author is grateful to the editor, Prof. Achmed Zayed, and the anonymous referee for their valuable comments that have substantively improved the manuscript.

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(1) In order to avoid ambiguity, the abbreviation TFT (THT) stands here for the Finite Fourier (Hankel) Transform, as the acronym FFT is usually used for the Fast Fourier Transform.

Tatiana Levitina

Institut Computational Mathematics, Technische Universitat Braunschweig

Braunschweig, D-38106, Germany

t.levitina@tu-bs.de

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Author: | Levitina, Tatiana |
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Publication: | Sampling Theory in Signal and Image Processing |

Article Type: | Report |

Geographic Code: | 4EUGE |

Date: | Jan 1, 2012 |

Words: | 6297 |

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