Prolate Spheroidal Wave Functions In q-Fourier Analysis

The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property [7]. They are an orthogonal basis of both L 2 (-1, 1) and the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. We prove that there are new systems possessing this property in q-Fourier analysis. In the following we discuss some properties of the q-Prolate spheroidal wave function using news developments and technics in q-Fourier analysis. In particular we prove that these functions forms an orthogonal basis of the q-Paley-Wiener space P W v q,a . Finally and as application we give a constructive q-sampling formula having as sampling points q n where n ∈ Z. In the end, we cit the reference [1], where the reproducing kernel for the q-Paley-Wiener space was already discussed, and the explicit formula for the kernel was given, similar to the formula in Remark 3. However, the paper [1] proceeds with a q-sampling theorem which extrapolates functions defined on the zeros of the q-Bessel function. These zeros are given in the following form

where 0 < ǫ n < 1, but it is not explicitly evaluated.

Throughout this paper we consider 0 < q < 1 and we adopt the standard conventional notations of [3]. We put

For complex z, let

Jackson’s q-integral in the interval [0, a] and in the interval [0, ∞[ are defined, respectively, by(see [4])

For v > -1, let L q,p,v be the space of even functions f defined on R q such that

The set L q,2,v is an Hilbert space with the inner product

We consider L q,v,a the space of function defined on [0, a] q which satisfies

and L v q,a the subspace of L q,2,v given by the natural embedding of L q,v,a in L q,2,v .

The normalized Hahn-Exton q-Bessel function of order v > -1 (see [6]) is defined by

(q, q) n (q v+1 , q) n z 2n .

It is an entire analytic function in z.

Proposition 1 For ℜ(v) > -1, a > 0 and y, z ∈ C{0} we have a 0 j v (yt, q 2 )j v (zt, q 2 )t 2v+1 d q t = 1 -q 1 -q 2v+2 a 2v+2 y 2 j v+1 (ay, q 2 )j v (aq -1 z, q 2 ) -z 2 j v+1 (az, q 2 )j v (aq -1 y, q 2 ) y 2 -z 2 .

Proof. See [5] (Proposition 1.

The following results in this section were proved in [2].

Proposition 2

The q-Bessel Fourier transform F q,v introduced in [2], [4] as follow

where

The q-Bessel translation operator is defined as follows:

In the following we tack q ∈ Q v where

The q-convolution product of both functions f, g ∈ L q,1,v is defined by

In the end we consider P W v q,a the q-Paley Wiener space

the set of q-bandlimited signal.

We introduce the q-analogue of the Prolate Spheroidal Wave Functions ψ i as the eigenfunction of the integral operator T v a acting on the Hilbert space L q,v,a as follows

then the sequence {ψ i } i∈N forme an orthogonal basis of the Hilbert space L q,v,a and any eigenvalue λ i is real.

Proposition 3 The sequence of eigenvalue {λ i } i∈N satisfying

Proof. The operator T v a is compact, then the spectrum is a countably infinite subset of R (T v a is symmetric) which has 0 as its only limit point. If we denote by Λ = {λ 0 , λ 1 , . . .}, the spectrum of T v a then we can write

To finish the proof, if suffice to prove that 0 / ∈ Λ. In fact if T v a ψ = 0 then F q,v ψ is an entire function which vanishes on [0, a] a . By the identity theorem for analytic functions, F q,v ψ = 0 everywhere and thus ψ = 0.

then K v a is an integral operator acting on the Hilbert space L q,v,a as follows

The function

The function ψ i initially defined on R q can be extended as an analytic function on C.

Proof. The result follows from the relation

and the fact that j v (., q 2 ) is an entire function.

The function ψ i belonging to the Paley-Wiener space P W v q,a Proof. Let

which implies that ψ i ∈ P W v q,a .

In the following we assume that

The sequence {ψ i } i∈N forme an orthonormal basis of P W v q,a .

Proof. The q-Bessel Fourier transform

define an isomorphism, and the sequence {φ i } i∈N form an orthogonal basis of the Hilbert space L v q,a , which lead to the result.

Proposition 6 Let k x : y → k(x, y),

and then

This finish the proof

Proof. In fact k x ∈ P W v q,a . Then

On the other hand

which prove the result.

Proof. In fact φ i , φ j = F q,v φ i , F q,v φ j = ψ i , ψ j ,

On the other hand, if i = j then

Moreover, φ i q,2,v = ψ i q,2,v = 1 which prove that φ i , φ j = δ ij . This leads to the result.

In order to be more precise about what it means for the energy of a q-bandlimited single f ∈ P W v q,a to be mainly concentrated on the interval [0, a] q , we consider the concentration index:

, whose values range from 0 to 1.

The maximum value of θ v a f is attained for f = ψ 0 and

Proof. With the Parseval equality

and the fact that

We get

which leads to the result.

Now let {µ n } n∈Z the sequence of eigenvalues of the operator T v b then we have

Proposition 8 The q-Pale

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