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 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.
Keywords:
Hankel transformation; fractional Hankel transformation; fractional Bessel wavelet transformation; Bessel function1 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 HankelHausdorff 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 onedimensional 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.
Applying the inverse fractional Hankel transformation of , we obtain
Proof
Since
using (8), we have
so that
Thus
□
and
2 Properties of a fractional Hankel transformation
Zemanian [[5], p.129] introduced a function space consisting of all complexvalued 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 complexvalued function on and for every choice of m and k of nonnegative 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]. □
The result can be easily shown by using Proposition 2.1(ii).
Proposition 2.2Let. Thensatisfies the following:
whereis a positive constant depending onμandθ.
(ii) From Proposition 2.1(ii), where , we have
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
□
3 Applications of the fractional Hankel transformation to generalized differential equations
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
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
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
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
Therefore,
□
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
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
□
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
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