Weinstein Gabor Transform and Applications

In this paper we consider Weinstein operator. We define and study the continuous Gabor transform associated with this operator. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. As applications, we obtain analogous of Heisenberg’s inequality for the generalized continuous Gabor transform. At the end we give the practical real inversion formula for the generalized continuous Gabor transform.


Introduction
We consider the Weinstein operator defined on where d is the Laplacian for the d-first variables and   the Bessel operator for the last variable, given by For , the operator   is the Laplace-Beltrami operator on the Riemanian space (cf. [1]).The Weinstein operator   2 L has several applications in pure and applied Mathematics especially in Fluid Mechanics (cf.[3]).
The harmonic analysis associated with the Weinstein operator is studied by Ben Nahia and Ben Salem (cf.[1,2]).In particular the authors have introduced and studied the generalized Fourier transform associated with the Weinstein operator.This transform is called the Weinstein transform.In this work we are interested to the Gabor transform associated with Weinstein operator.
Time-Frequency analysis plays a central role in signal analysis.Since years ago, it has been recognized that the global Fourier transform of a long time signal has a little practical value to method is preferred to the classical Fourier method, whenever the time dependence of the analyzed signal is of the same importance as its frequency dependence.
However, there exist strict limits to the maximal Time-Frequency resolution of this transform, similar to Heisenberg's uncertainty principles in the Fourier analysis.
In fact, Dennis Gabor [4] was the first to introduce the Gabor transform, in which he uses translations and modulations of a single Gaussian to represent one dimensional signal.Other names for this transform used in literature, are: short time Fourier transform, Weyl-Heisenberg transform, windowed Fourier transform.
In this paper, we are interested a generalized Gabor transform associated for the Weinstein transform.More precisely, we give here general reconstruction formulas and we give many applications.In the classical case the Gabor transform is very fundamental and has many applications to Mathematical Sciences.
The paper is organized as follows.In Section 2, we recall the main results about the harmonic analysis related to the Weinstein operator.In Section 3, we introduce the analogue of the continuous Gabor transform associated with the Weinstein operator and we give some harmonic properties for it (Plancheral formula,  inverse formula, weak uncertainty for it).The Section 4 is devoted to prove the analogous of Heisenberg's inequality for the generalized continuous Gabor transform.In Section 5 using the kernel reproducing theory given by Saitoh [5] we study the problem of approximative concentration.In the last section we give a practical real inversion formulas and extremal function for the Weinstein-Gabor transform.

 
briefly overview the Weinstein operator and related harmonic analysis.Main references are [1,2].

  
In the following we denote by .
, , , , even with respect to the last variable.
the space of functions of class p C on , even with respect to the last variable.
, even with respect to the last variable.
 the Schwartz space of rapidly decreasing functions on , even with respect to the last variable.

 
the space of -functions on   which are of compact support, even with respect to the last variable.
We consider the Weinstein operator   defined by , , , where is the Laplace operator on , and given by The Weinstein kernel is given by where is the normalized Bessel function.The Weinstein kernel satisfies the following properties: 1) For all , we have , , 2) For all , and where


We denote by the space of measurable functions on d where   is the measure on given by Some basic properties of this transform are the following: , for all .
The Weinstein transform W is a topological isomorphism from onto itself and for all f in   In particular, the Weinstein transform f f   can be uniquely extended to an isometric isomorphism from   By using the Weinstein kernel, we can also define a generalized translation.For a function is defined by the following relation: For example, for , we see that By using the generalized translation, we define the generalized convolution product f g of functions as follows: , f g This convolution is commutative and associative and satisfies the following propositions: If and , then and , and in this case we have An immediate consequence of Proposition 3 and the Plancherel formula that will be used in the next section is the following.
Proposition 4 Let f and g be in .Then, we have where both sides are finite or infinite.

The Continuous Weinstein Gabor Transform
Notations.We denote by:    the space of measurable functions f on with respect to the measure : ess sup , .

 Definition 1. For any function g in and any
    , we define the modulation of g by v as: where  y  ,    , are the Weinstein translation operators given by (13).  .
We consider the family we define its continuous Weinstein Gabor transform by which can also be written in the form , where in For proof this theorem we need the following Lemmas.
Lemma 1.Let g be as above.For any positive integer n define the two functions and .
. Then Proof.Using the Cauchy-Schwartz inequality we obtain 3 Therefore by Fubini theorem, the inversion theorem, the Plancherel formula and Proposition  is easily checked.Finally, using Fubinis theorem we obtain Let g be as above.For any positive integer n the function .
Proof.We have .
On the follow we justify the use of Fubinis theorem in the last sequence of equalities observe that It follows from Proposition 3 and Lem By this, the Plancherel formula, the hat 1 n H  em, it follows that fact t pointwise as n   , and the dominated convergence theor Proof.Using relation (18), Fubini's theorem and Pl , we 2, ancherel's formula for the Weinstein transform have e classical case, the continuous Weinstein Ga-bor transform preserves the orthogonality re ever, we have the following result.
Then, for all f, h in As in th e continuous Weinstein Gabor transform.
Proposition 7. Let g be in


In what follows, we show the weak uncertainty princeple for th where  : Proof.From the relation (24) we deduce that Proof.Using Proposition 5 and Pro osition 6 the result follows by applying the Riesz-Thori nterpolation theo-

Uncertainty Principles of Heisenberg Type
In equality for the generalized continuous Gabo ansform.

Proposition 9. (Uncertainty principle of Heisenberg type for
). n this section we will to prove the Heisenberg in r tr


, the following inequality holds lt by combining the Heisenbe sical Fourier t sform and Fo m 2. (Uncert rinciples berg x Proof.We obtain the resu rg inequalities for the cl Type for g  ). as ran urier-Bessel transform.
Let g be in the following inequality holds Theore ainty p of Heisen Let us assume the non-trivial case that both ingrals on the left hand side of (28) are finite.Fixing d , .
arbitrary, Heisenberg's inequality for einstein transform gives that ., d Integrating over  and using Cauchy Schwartz inequality we obtain .
., d d d (Reproducing Kernel).Let g be in


This proves the result.
t space in a reproducing kernel Hilber 2
Proof.We have Using the relation (26), we obtain On the other hand using Proposition 3, one can easily


. Therefore, the result is obtained.llowing theorem, we will show that the portion of the continuous Weinstein Gabor transform lying utside some sufficiently small set of finite measure a f the cont the following U g U g Then, for all f in Proof.From the definition of U P and As , , , , , and its Hilbert-Schmidt norm

Practical Real Inversion Formulas for  g
Thus, from the relations (34), (3 and (36) we obtain the re   .We define the space provided with the inner product and the norm , is a Hilbert space.
Proposition 10.Let g be a function in ., ., the subspace of function supported in the subset

3 .
We proceed as[6]  we obtain the result.Definition Let g be a function in L rm associated to the inner product is defined by: we prove the following results.tion11 et g be in .