The Continuous Wavelet Transform for a q-Bessel Type Operator

on harmonic analysis tools with respect to the  we study some definitions and properties of q-Bessel continuous wavelet transform. We also explore generalized q-Bessel Fourier transform and convolution product on associated with the operator  and finally a new continuous wavelet transform associated with q-Bessel operator is constructed and investigated.

where, We can write ( ) ( ) (1.3) From above equations, we can say that wavelet transform of the function f on R is an integral transform and the dilated translate of  is the kernel.
We can also express wavelet transform as the convolution: Where, ( ) ( ).
Since there is a special type of convolution for every integral transform, therefore one can define wavelet transform with respect to a integral transform using associated convolution.
The concept of wavelet is a collection of function derived from a single function called mother wavelet, after that by applying the two operators known as translation and dilation we get a new type of continuous wavelet transform.
Unlike the elementary functions such as trigonometric, exponential etc the Bessel wavelets are related to special functions and Jachkson introduced the concept of q-analysis at the beginning of the twentieth century.We have arranged this paper as follows: In section 2, we will review briefly the basics of q-Bessel Fourier transform, here we recall notations, some definitions of q-Bessel Fourier and Inverse Fourier transform and the preposition associated with other operators and convolution product.In section 3, some results of harmonic analysis with respect to q-Bessel operator for the generalized q-Bessel transform is collected and the definition and properties of convolution product is also discussed.To extend the classical theory of wavelets to the differential operator , q   is the actual aim of this work.
We define a generalized wavelet, which satisfy the below admissibility condition ,, 0 0.
Where , q F   denotes the generalized q-Bessel Fourier transform related to operator given by , . ; Starting with a single generalized wavelet g, a family of generalized wavelets is constructed by putting ,, 0 ,.
In section 4, we develop a relationship between the generalized wavelet transforms and q-Bessel continuous wavelet transforms.Such a relationship helps us to build certain formulas for the generalized q-Bessel continuous wavelet transform (CWT).
In Section 5, we study the intertwining operator q  to establish the continuous generalized q- Bessel wavelet transform in form of classical one.As a result, we got a new inversion formulas for dual operator

Preliminaries
In the present section we recapitulate some facts about harmonic analysis related to the q-Bessel operator.We cite here, as briefly as possible, only those properties actually required for the discussion.
Throughout this section assume 1/ 2

 −
. Let the space ,, qp L  , 1 p    denote the sets of real functions on q + for which ( ) , The q-Bessel Fourier transform , ,, where j  is normalized q-Bessel function.
for almost all The inverse transform is given by ,, The q-Bessel translation operators , , 0, ,, 0 , , , where D x y z c j xs q j ys q j zs q s d s The convolution product of q-Bessel for two functions , fg is defined as ,, 0 , 0. . ,. .
be a q-Bessel wavelet of order  .Then continuous q-Bessel wavelet transform is defined as follows ,, (2.12) The q-Bessel continuous wavelet transform has been investigated in detail in [4] from which we see the following basis properties.

Harmonic Analysis Associated with  and Generalized Fourier Transform
Let M be the map defined by .

Generalized Convolution Product
Definition 4.1 The generalized translation operator ,, q x n T  is define by the relation where ,2 qn c  + is given by (1.6).
From by (4.1) we have where ,2 , , 00 .q x n q q x n q T f y g y y d y f y T y y d y Proof.(i) By (4.1) and Theorem 2.2(i) we have ,1, , . , , , ,.
. n n n q x n q q x q n n n q x q n nn q x q n q x n q q x n q

T f y g y y d y x y M f y M g y y d y x y M f y M g y y d y y M f y xy M g y y d y y M f y T g y y d y f y T g y y d y
.
This concludes the proof.
Remark 5.1 (i) For n=0, q  reduces to q-Riemann Liouville integral transform of order  given by ( )( ) (iii) From Theorem 3.1(i) and (5.1) we have (  for a differential function :  reduces to q-Weyl integral transform of order  given by ( )( ) ( ) ( ) ( ) , where , qC F is the q-cosine Fourier transform given by ( )( ) where * is the convolution product defined by  is a q-generalized translation given in details in [5]. )

=
This achieves the proof.6. Generalized Wavelets Definition 6.1 A generalized q-Bessel wavelet is a function is a generalized q-Bessel wavelet if and only if, Mg − is a q-Bessel wavelet of order 2n  + , and we have ( )( ) where a g is given in (2.12) and , qb T  are the generalized translation operators defined by (4.1).Theorem 6.1 For all q a +  and   Proof.Using (2.11), (4.1) and ( 6.3) we have , be a generalized a q-Bessel wavelet.Then for a function , the continuous generalized a q-Bessel wavelet transform by ( It can also be written in the form ( )( ) ( ) where # q is the generalized convolution product given by (4.2).Theorem 6.2 We have ( Proof.From (2.10), (6.4) and (6.5) we deduce that , , , 0 Proof.By (6.2) and Theorem 2.1(i) we have , Proof.By (6.2), (6.4), (6.7) and theorem 2.1(ii) we have . , Proof.By (6.2), (6.4), (6.7 we have the result shows from theorem 2.1(iii). involving generalized wavelets, we have to establish some preliminary lemmas.

Inversion of the Intertwining Operator
.
The continuous wavelet transform on  ) is the normalized Bessel function with index 2n

1
The generalized q-Bessel Fourier transform is defined for a function The inverse of this transform is given by 3.6) and theorem 2.1 (iii) we have ( ) the Bessel translation operators of order 2n  + .Definition 4.2 Define the generalized convolution product of two functions f and g on [0, )  by By (3.1), (3.6), (4.1) and Theorem 2.2(ii) we have By (4.3) and Theorem 2.2(iv) we have


Through the Generalized Wavelet TransformTo obtain inversion formulas or t q


From the Plancherel theorem for the cosine transform, it follows that

g
be as in Lemma 7.2.Then we have the following inversion formulas for t q The author(s) declare that there are no conflicts of interest regarding the publication of this paper. .