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 nth 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.

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 ℝ 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

and its inversion formula is given by

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:

where the kernel

and

The inversion formula of (3) is given by

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:

where the fractional Hankel translation of the function is defined by

and

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

for .

Applying the inverse fractional Hankel transformation of , we obtain

Lemma 1.1If, then

Proof

Since

using (8), we have

so that

Thus

 □

Remark 1.1 If , then

and

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

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

where

and

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

Proposition 2.1Letbe the kernel of the fractional Hankel transformation. Then

whereand is known as a fractional Bessel operator with parameterθ.

Proof See [13]. □

Example 2.1, .

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

Proposition 2.2Let. Thensatisfies the following:

whereis a positive constant depending onμandθ.

Proof (i) Clearly, .

So, .

(ii) From Proposition 2.1(ii), where , we have

although as for every .

(iii) Let , consider

and

So,

This proves that is continuous in . □

Proposition 2.3 (Parseval’s relation)

Ifanddenote the fractional Hankel transformations ofandrespectively, then

and

Proof

We have

If , then

 □

We consider the generalized nth order linear nonhomogeneous ordinary differential equation

where L is the generalized nth 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,

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

Let be the FrHT of order zero of with respect to the variable x. Then, by definition,

where is the kernel of FrHT of order zero.

Taking the FrHT of order zero of (17), we get

where , whose solution is

Taking the FrHT of order zero of (18), we have

Condition (22) is satisfied if we have .

Therefore, from (21)

Taking the FrHT of order zero of (19), we have

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

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 4.1If, then

Proof

We have

Now,

Therefore,

Thus,

 □

Theorem 4.1Let. Then the continuous fractional Bessel wavelet transformationis defined onfby

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 4.2Ifandare two wavelets andanddenote the continuous fractional Bessel wavelet transformations ofrespectively, then

where

Proof

We have

Now,

 □

Theorem 4.3Ifψis a wavelet andandare the continuous fractional Bessel wavelet transformations ofrespectively, 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 4.4Let. Thenfcan be reconstructed by the formula

Proof For any , we have

Therefore,

 □

Theorem 4.5If, then

where

Proof

Using Theorem 4.1 and Theorem 4.2, we have

This completes the proof of the theorem. □

Theorem 4.6Ifis a Bessel wavelet andfis a bounded integrable function, then the convolutionis 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 4.7Ifandis 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 .

(ii) We have

Therefore, by the Hölder inequality, we have

 □

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

Dedicated to Professor Hari M Srivastava.

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