PaleyWiener Theorem for the Weinstein Transform and applications
Abstract.
In this paper our aim is to establish the PaleyWiener Theorems for the Weinstein Transform. Furthermore, some applications are presents, in particular some properties for the generalized translation operator associated with the Weinstein operator are proved.
keywords: Weinstein transform, PaleyWiener Theorem.
Mathematics Subject Classification (2010): 42B10, 44A15, 44A20, 30G35.
1. Introduction
PaleyWiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform [7]. For example, the classical PaleyWiener theorem (see [9]) stated that states that a necessary and sufficient condition for a squareintegrable function to be extendable to an entire function in the complex plane with an exponential type bound
if is bandlimited, i.e. the Fourier transform has compact support. Higher dimensional extensions of the PaleyWiener theorem have been studied. There are plenty of PaleyWiener type theorems since there are many kinds of bound for decay rates of functions and many types of characterizations of smoothness. In this regard a wide number of papers have been devoted to the extension of the theory on many other transforms and different classes of functions, for example, the Mellin transform [10], Hankel transform [4, 11], Jacobi transform [12] and CliffordFourier transform [8, 13] .
Since the Weinstein transform are natural generalizations of the Fourier transforms, it is natural to ask whether such a representation for entire functions is possible in this case also. The aim of this paper is to obtain an analogue of the PaleyWiener theorem for Weinstein transforms. As applications of this results, some properties for the generalized translation operator associated with the Weinstein operator are established.
2. Harmonic analysis Associated with the Weinstein Operator
In order to set up basic and standard notation we briefly overview the Weinstein operator and related harmonic analysis. Main references are [1, 2].
In the following we denote by

.



, the space of continuous functions on even with respect to the last variable.

, the space of the functions, even with respect to the last variable, and rapidly decreasing together with their derivatives.

the space of measurable functions on such that
where
(1) 
the set of homogeneous polynomials on of degree , even with respect to the last variable.

the Wiener algebra space.

We consider the Weinstein operator defined on by
(2) 
where is the Laplacian operator for the first variables and is the Bessel operator for the last variable defined on by
The Weinstein operator have remarkable applications in diffrerent branches of mathematics. For instance, they play a role in Fluid Mechanics [3].
2.1. The eigenfunction of the Weinstein operator
For all , the system
(3) 
has a unique solution on , denoted by and given by
(4) 
where and is is the normalized Bessel function of index defined by
The function has a unique extension to , and satisfied the following properties:
Proposition 1.
i). For all we have
(5) 
ii). For all we have
(6) 
iii). For all we get
(7) 
vi). For all and we have
(8) 
where and In particular, for all , we have
(9) 
2.2. The Weinstein transform
Definition 1.
Proposition 2.

For all , the function is continuous on and we have
(11) 
The Weinstein transform is a topological isomorphism from onto itself. The inverse transform is given by
(12) where
(13) 
Parseval formula: For all , we have
(14) 
Plancherel formula: For all , we have
(15) 
Inversion formula: If , then
(16)
2.3. The translation operator associated with the Weinstein operator
Definition 2.
The translation operator associated with the Weinstein operator , is defined for a continuous function on which is even with respect to the last variable and for all by
By using the Weinstein kernel, we can also define a generalized translation, for a function and the generalized translation is defined by the following relation
(17) 
The following proposition summarizes some properties of the Weinstein translation operator.
Proposition 3.
The translation operator satisfies the following properties:
i). For , we have for all
ii). Let and . Then belongs to and we have
Note that the is contained in the intersection of and and hence is a subspace of For we have
(18) 
By using the generalized translation, we define the generalized convolution product of the functions as follows
(19) 
This convolution is commutative and associative, and it satisfies the following properties:
Proposition 4.
i). For all (resp. ), then (resp. ) and we have
(20) 
ii). Let such that
(21) 
2.4. Heat functions related to the Weinstein operator
The generalized heat kernel associated with the Weinstein operator is given by
(22) 
which is a solution of the generalized heat equation:
and satisfied the following properties ( see [6]):
Proposition 5.
i). For , the function is nonnegative.
ii). For , we have
(23) 
iii). For , we have
(24) 
iv). For , we have
(25) 
3. PaleyWiener Theorem for the Weinstein Transform
In this section we prove a sharp PaleyWiener theorem for the Weinstein transform and study its consequences. We suppose that and For a nonnegative integer , we put
which is called the space of generalized spherical harmonics of degree We fix a and define the Weinstein coefficients of in the angular variable by
(26) 
with . Then the Weinstein spherical harmonic coefficients of are given by
(27) 
Theorem 1.
Let and be a positive number. Then is supported in if and only if the Weinstein spherical harmonic coefficients of extends to an entire function of satisfying the estimate
Proof.
In [5] (see proof of Proposition 5), the authors proved that
where as defined in (26). Thus, is the Hankel transform of order of the function So, PaleyWiener Theorem for the Hankel transform ( see [4]) and (26) completes the proof.
Theorem 2.
A be supported in if and only if the function extends to an entire function of which satisfies
(28) 
Proof.
Since the Weinstein kernel is an entire function in and satisfied
On the other hand, by the definition of the Weinstein transform we have
where
Conversely, assume that the function is an entire function of and satisfied (28), thus implies that the function
is an entire function of exponential type , from which the converse follows from the Theorem 1.
Corollary 1.
Let be supported in . Then is supported in .
Proof.
Let Then Thus
i.e extends to as an entire function of type . Consequently, from the previous Theorem we conclude that is supported in .
Corollary 2.
Let . If is supported in , then the following inequality
(29) 
holds for all .
Proof.
Inversion formula (16) and (18) yields that
(30) 
Combining (8) and (30) and using the mean value theorem we get
As is supported in and is supported in we can restrict the integration domain above to A short calculation gives the desired result.
Corollary 3.
If then
(31) 
Proof.
From the fact and we get
By Fubini’s Theorem and the equality (24) we obtain
(32) 
For a given we choose such that Now, we can write in the following form
So, the triangle inequality and (32) leads to
On the other hand, by using (23) we have
Thus implies then
If is supported in , by Corollary 2 we get
which can be made smaller than by choosing small. So, the proof of Corollary 3 is completes.
Corollary 4.
For , then for almost
(33) 
Proof.
Let . Using (14), (17) and (25) we have
This extends to since the convolution operator extends to as a bounded operator by the inequality (32). Letting applying Corollary 3 to
the lefthand side and the dominant convergence theorem to the righthand side, we see
that the inversion formula follows almost everywhere.
Corollary 5.
Suppose that is open. Suppose, further that are pairwise distinct and If for all then for all
Proof.
By successive analytic continuation in each coordinate we can derive that the assumption actually means that for all . Let be a test function. Then
for all . Since and The Weinstein transform is a topological isomorphism from onto itself, we have
(34) 
for all . Now take to be compactly supported and support contained in the ball around zero with radius then for is compactly supported and support contained in the ball around zero with radius for all by means of Corollary 1. Now, suppose that thus implies that
So, by again using (34) we obtain that for all and consequently
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