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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

The continuous fractional Bessel wavelet transformation

Akhilesh Prasad1, Ashutosh Mahato1, Vishal Kumar Singh1 and Madan Mohan Dixit2*

Author affiliations

1 Department of Applied Mathematics, Indian School of Mines, Dhanbad, 826004, India

2 Department of Mathematics, NERIST, Nirjuli, India

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Citation and License

Boundary Value Problems 2013, 2013:40  doi:10.1186/1687-2770-2013-40


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/40


Received:17 November 2012
Accepted:25 January 2013
Published:27 February 2013

© 2013 Prasad et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M1">View MathML</a> denote the class of measurable functions of ϕ on ℝ such that the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M2">View MathML</a> is finite. Also, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M3">View MathML</a> be a collection of almost everywhere bounded functions, hence endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M4">View MathML</a>

The Hankel transformation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M5">View MathML</a>, [5] of a conventional function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M7">View MathML</a> is usually defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M8">View MathML</a>

(1)

and its inversion formula is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M9">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M10">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M13">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M14">View MathML</a>

(3)

where the kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M15">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M16">View MathML</a>

The inversion formula of (3) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M17">View MathML</a>

(4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M18">View MathML</a>

and

We assume that throughout this paper <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M21">View MathML</a>.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M22">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M23">View MathML</a>

for all a, b and θ as above.

As per [2,12], we defined the fractional Hankel convolution of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M24">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M25">View MathML</a>

(5)

where the fractional Hankel translation of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M6">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M27">View MathML</a>

(6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M28">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M29">View MathML</a> denotes the area of a triangle with sides x, y, z of such a triangle exists and zero otherwise. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M30">View MathML</a> and is symmetric in x, y, z.

Now, setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M31">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M32">View MathML</a>

(8)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M33">View MathML</a>.

Applying the inverse fractional Hankel transformation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M34">View MathML</a>, we obtain

Lemma 1.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M36">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M37">View MathML</a>

Proof

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M38">View MathML</a>

using (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M39">View MathML</a>

(9)

so that

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M41">View MathML</a>

 □

Remark 1.1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M36">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M43">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M44">View MathML</a>

2 Properties of a fractional Hankel transformation

Zemanian [[5], p.129] introduced a function space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M45">View MathML</a> consisting of all complex-valued infinitely differentiable function φ defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M7">View MathML</a>, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M47">View MathML</a>

(10)

Definition 2.1 (Test function space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M48">View MathML</a>)

The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M48">View MathML</a> is defined as follows: φ is a member of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M48">View MathML</a> if and only if it is a complex-valued <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M51">View MathML</a>-function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M52">View MathML</a> and for every choice of m and k of non-negative integers, it satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M53">View MathML</a>

(11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M54">View MathML</a>

(12)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M55">View MathML</a>

where the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M56">View MathML</a> depend only on μ and parameter θ. On <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M48">View MathML</a>, we consider the topology generated by the family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M58">View MathML</a> of seminorms.

Proposition 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M59">View MathML</a>be the kernel of the fractional Hankel transformation. Then

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M61">View MathML</a>and is known as a fractional Bessel operator with parameterθ.

Proof See [13]. □

Example 2.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M63">View MathML</a>.

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

Proposition 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M64">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M65">View MathML</a>satisfies the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M66">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M67">View MathML</a>is a positive constant depending onμandθ.

Proof (i) Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M68">View MathML</a>.

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M69">View MathML</a>.

(ii) From Proposition 2.1(ii), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M70">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M71">View MathML</a>

although <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M72">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M73">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M64">View MathML</a>.

(iii) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M75">View MathML</a>, consider

and

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M78">View MathML</a>

This proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M79">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M52">View MathML</a>. □

Proposition 2.3 (Parseval’s relation)

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M81">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M82">View MathML</a>denote the fractional Hankel transformations of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M11">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M84">View MathML</a>respectively, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M85">View MathML</a>

(13)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M86">View MathML</a>

(14)

Proof

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M87">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M88">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M89">View MathML</a>

 □

3 Applications of the fractional Hankel transformation to generalized differential equations

We consider the generalized nth order linear nonhomogeneous ordinary differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M90">View MathML</a>

(15)

where L is the generalized nth order differential operator given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M91">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M92">View MathML</a> are constants and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M93">View MathML</a> is as given in Proposition 2.1.

Applying FrHT to both sides of equation (15), we have

and equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M95">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M96">View MathML</a>

(16)

Now, an application of the inverse FrHT gives the solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M97">View MathML</a>

Example 3.1 Let us consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M98">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M99">View MathML</a>

Example 3.2 Using the FrHT, we investigate the solution of the generalized differential equation

(17)

(18)

(19)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M103">View MathML</a> be the FrHT of order zero of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M104">View MathML</a> with respect to the variable x. Then, by definition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M105">View MathML</a>

(20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M106">View MathML</a> is the kernel of FrHT of order zero.

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

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M108">View MathML</a>, whose solution is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M109">View MathML</a>

(21)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M110">View MathML</a>

(22)

Condition (22) is satisfied if we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M111">View MathML</a>.

Therefore, from (21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M112">View MathML</a>

(23)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M113">View MathML</a>

(24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M114">View MathML</a> is the FrHT of zero order of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M115">View MathML</a>.

Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M116">View MathML</a> in (23) and using (24), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M117">View MathML</a>.

Hence (23) reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M118">View MathML</a>

Applying the inversion formula, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M119">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M120">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M121">View MathML</a> which satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M122">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M123">View MathML</a> is called the admissibility condition of the fractional Bessel wavelet and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M124">View MathML</a> is the fractional Hankel transformation of ψ. The fractional Bessel wavelets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M125">View MathML</a> are generated from one single function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M121">View MathML</a> by dilation and translation with parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M127">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M128">View MathML</a> respectively by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M129">View MathML</a>

Lemma 4.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M130">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M131">View MathML</a>

Proof

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M132">View MathML</a>

Now,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M133">View MathML</a>

Therefore,

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M135">View MathML</a>

 □

Theorem 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M136">View MathML</a>. Then the continuous fractional Bessel wavelet transformation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M137">View MathML</a>is defined onfby

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M138">View MathML</a>

Proof

We have

by putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M140">View MathML</a>, then the continuous fractional Bessel wavelet transformation can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M141">View MathML</a>

This means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M142">View MathML</a>

 □

Remark 4.1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M143">View MathML</a> is a homogeneous function of degree n, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M144">View MathML</a>

Theorem 4.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M145">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M146">View MathML</a>are two wavelets and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M147">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M148">View MathML</a>denote the continuous fractional Bessel wavelet transformations of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M149">View MathML</a>respectively, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M150">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M151">View MathML</a>

Proof

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M152">View MathML</a>

Now,

 □

Theorem 4.3Ifψis a wavelet and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M154">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M155">View MathML</a>are the continuous fractional Bessel wavelet transformations of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M156">View MathML</a>respectively, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M157">View MathML</a>

Proof The proof of Theorem 4.3 can be easily deduced by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M158">View MathML</a> in Theorem 4.2. □

Remark 4.2 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M159">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M158">View MathML</a>, then from Theorem 4.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M161">View MathML</a>

Theorem 4.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M143">View MathML</a>. Thenfcan be reconstructed by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M163">View MathML</a>

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M164">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M165">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M166">View MathML</a>

 □

Theorem 4.5If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M121">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M168">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M169">View MathML</a>

Proof

Using Theorem 4.1 and Theorem 4.2, we have

This completes the proof of the theorem. □

Theorem 4.6If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M130">View MathML</a>is a Bessel wavelet andfis a bounded integrable function, then the convolution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M172">View MathML</a>is a fractional Bessel wavelet, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M173">View MathML</a>

Proof

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M174">View MathML</a>

Therefore,

This implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M176">View MathML</a>

We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M177">View MathML</a>. Moreover,

Thus, the convolution function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M172">View MathML</a> is a fractional Bessel wavelet. □

Theorem 4.7If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M180">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M181">View MathML</a>is the continuous fractional Bessel wavelet transformation, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M182">View MathML</a>

Proof (i) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M183">View MathML</a> be an arbitrary but fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M184">View MathML</a>. Then, by the Hölder inequality,

Since

and

by the dominated convergence theorem and the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M188">View MathML</a> in the variable b and a, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M189">View MathML</a>

This proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M181">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M184">View MathML</a>.

(ii) We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M192">View MathML</a>

Therefore, by the Hölder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/40/mathml/M193">View MathML</a>

 □

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.

References

  1. Pathak, RS, Dixit, MM: Continuous and discrete Bessel wavelet transforms. J. Comput. Appl. Math.. 160(1-2), 241–250 (2003). Publisher Full Text OpenURL

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