## Boundary Value Problems

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# The continuous fractional Bessel wavelet transformation

Boundary Value Problems20132013:40

DOI: 10.1186/1687-2770-2013-40

Accepted: 25 January 2013

Published: 27 February 2013

## Abstract

The main objective of this paper is to study the fractional Hankel transformation and the continuous fractional Bessel wavelet transformation and some of their basic properties. Applications of the fractional Hankel transformation (FrHT) in solving generalized n th order linear nonhomogeneous ordinary differential equations are given. The continuous fractional Bessel wavelet transformation, its inversion formula and Parseval’s relation for the continuous fractional Bessel wavelet transformation are also studied.

MSC:46F12, 26A33.

### Keywords

Hankel transformation fractional Hankel transformation fractional Bessel wavelet transformation Bessel function

## 1 Introduction

Pathak and Dixit [1] introduced continuous and discrete Bessel wavelet transformations and studied their properties by exploiting the Hankel convolution of Haimo [2] and Hirschman [3]. Upadhyay et al. [4] studied the continuous Bessel wavelet transformation associated with the Hankel-Hausdorff operator.

Let denote the class of measurable functions of ϕ on such that the integral is finite. Also, let be a collection of almost everywhere bounded functions, hence endowed with the norm
The Hankel transformation , [5] of a conventional function , is usually defined by
(1)
and its inversion formula is given by
(2)

where is the Bessel function of the first kind of order μ.

The fractional Hankel transformation is the generalization of the conventional Hankel transformation in the fractional order with parameter θ and is effectively used in the design of lens, analysis of laser cavity study of wave propagation in quadratic refractive index medium when the system is axially symmetric. The earliest work on the fractional Hankel transformation was published by Namias in 1980 [6]. Recently, it has become of importance in various applications in optics [7, 8]. Kerr [9] has developed a theory of fractional power of Hankel transforms in Zemanian spaces. We define a one-dimensional fractional Hankel transformation (FrHT) with parameter θ of for and as follows:
(3)
where the kernel
and
The inversion formula of (3) is given by
(4)
where
and

We assume that throughout this paper , .

From [10], wavelets as a family of functions constructed from translation and dilation of a single function ψ are called the mother wavelet defined by
where a is called the scaling parameter which measures the degree of compression or scale and b is a translation parameter which determines the time location of the wavelet. Shi et al. [11] defined the fractional mother wavelet as

for all a, b and θ as above.

As per [2, 12], we defined the fractional Hankel convolution of functions as follows:
(5)
where the fractional Hankel translation of the function is defined by
(6)
and
(7)

where denotes the area of a triangle with sides x, y, z of such a triangle exists and zero otherwise. Clearly, and is symmetric in x, y, z.

Now, setting , we have
(8)

for .

Applying the inverse fractional Hankel transformation of , we obtain
Lemma 1.1 If , then

Proof

Since
using (8), we have
(9)
so that
Thus

□

Remark 1.1 If , then
and

## 2 Properties of a fractional Hankel transformation

Zemanian [[5], p.129] introduced a function space consisting of all complex-valued infinitely differentiable function φ defined on , satisfying
(10)

Definition 2.1 (Test function space )

The space is defined as follows: φ is a member of if and only if it is a complex-valued -function on and for every choice of m and k of non-negative integers, it satisfies
(11)
where
(12)
and

where the constants depend only on μ and parameter θ. On , we consider the topology generated by the family of seminorms.

Proposition 2.1 Let be the kernel of the fractional Hankel transformation. Then

where and is known as a fractional Bessel operator with parameter θ.

Proof See [13]. □

Example 2.1 , .

The result can be easily shown by using Proposition 2.1(ii).

Proposition 2.2 Let . Then satisfies the following:

where is a positive constant depending on μ and θ.

Proof (i) Clearly, .

So, .
1. (ii)
From Proposition 2.1(ii), where , we have

although as for every .
1. (iii)
Let , consider

and
So,

This proves that is continuous in . □

Proposition 2.3 (Parseval’s relation)

If and denote the fractional Hankel transformations of and respectively, then
(13)
and
(14)

Proof

We have
If , then

□

## 3 Applications of the fractional Hankel transformation to generalized differential equations

We consider the generalized n th order linear nonhomogeneous ordinary differential equation
(15)
where L is the generalized n th order differential operator given by

where are constants and is as given in Proposition 2.1.

Applying FrHT to both sides of equation (15), we have
and equivalently,
Therefore,
(16)
Now, an application of the inverse FrHT gives the solution
Example 3.1 Let us consider . Then we have
Example 3.2 Using the FrHT, we investigate the solution of the generalized differential equation
(17)
(18)
(19)
Let be the FrHT of order zero of with respect to the variable x. Then, by definition,
(20)

where is the kernel of FrHT of order zero.

Taking the FrHT of order zero of (17), we get
where , whose solution is
(21)
Taking the FrHT of order zero of (18), we have
(22)

Condition (22) is satisfied if we have .

Therefore, from (21)
(23)
Taking the FrHT of order zero of (19), we have
(24)

where is the FrHT of zero order of .

Putting in (23) and using (24), we get .

Hence (23) reduces to
Applying the inversion formula, we have

## 4 The continuous fractional Bessel wavelet transformation

The continuous fractional Bessel wavelet transformation (CFrBWT) is a generalization of the ordinary continuous Bessel wavelet transformation (CBWT) with parameter θ, that is, CBWT is a special case of CFrBWT with parameter . In this section, we define the continuous fractional Bessel wavelet transformation and study some of its properties using the theory of fractional Hankel convolution (5) corresponding to [10].

A fractional Bessel wavelet is a function which satisfies the condition
where is called the admissibility condition of the fractional Bessel wavelet and is the fractional Hankel transformation of ψ. The fractional Bessel wavelets are generated from one single function by dilation and translation with parameters and respectively by
Lemma 4.1 If , then

Proof

We have
Now,
Therefore,
Thus,

□

Theorem 4.1 Let . Then the continuous fractional Bessel wavelet transformation is defined on f by

Proof

We have
by putting , then the continuous fractional Bessel wavelet transformation can be written as
This means that

□

Remark 4.1 If is a homogeneous function of degree n, then
Theorem 4.2 If and are two wavelets and and denote the continuous fractional Bessel wavelet transformations of respectively, then
where

Proof

We have
Now,

□

Theorem 4.3 If ψ is a wavelet and and are the continuous fractional Bessel wavelet transformations of respectively, then

Proof The proof of Theorem 4.3 can be easily deduced by setting in Theorem 4.2. □

Remark 4.2 If and , then from Theorem 4.3, we have
Theorem 4.4 Let . Then f can be reconstructed by the formula
Proof For any , we have
Therefore,

□

Theorem 4.5 If , then
where

Proof

Using Theorem 4.1 and Theorem 4.2, we have

This completes the proof of the theorem. □

Theorem 4.6 If is a Bessel wavelet and f is a bounded integrable function, then the convolution is a fractional Bessel wavelet, where

Proof

We have
Therefore,
This implies that
We have . Moreover,

Thus, the convolution function is a fractional Bessel wavelet. □

Theorem 4.7 If and is the continuous fractional Bessel wavelet transformation, then
Proof (i) Let be an arbitrary but fixed point in . Then, by the Hölder inequality,
Since
and
by the dominated convergence theorem and the continuity of in the variable b and a, we have
This proves that is continuous on .
1. (ii)
We have

Therefore, by the Hölder inequality, we have

□

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

## Authors’ Affiliations

(1)
Department of Applied Mathematics, Indian School of Mines
(2)
Department of Mathematics, NERIST

## References

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