Open Access

The continuous fractional Bessel wavelet transformation

  • Akhilesh Prasad1,
  • Ashutosh Mahato1,
  • Vishal Kumar Singh1 and
  • Madan Mohan Dixit2Email author
Boundary Value Problems20132013:40

DOI: 10.1186/1687-2770-2013-40

Received: 17 November 2012

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 L p ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq1_HTML.gif denote the class of measurable functions of ϕ on such that the integral R | ϕ ( x ) | p d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq2_HTML.gif is finite. Also, let L ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq3_HTML.gif be a collection of almost everywhere bounded functions, hence endowed with the norm
ϕ L p = { ( R | ϕ ( x ) | p d x ) 1 / p , 1 p < , ess sup x R | ϕ ( x ) | , p = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equa_HTML.gif
The Hankel transformation h μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq4_HTML.gif, [5] of a conventional function φ L 1 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq5_HTML.gif, R + = ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq6_HTML.gif is usually defined by
φ ˆ ( y ) = ( h μ φ ) ( y ) = 0 ( x y ) 1 2 J μ ( x y ) φ ( x ) d x , x R + , μ 1 / 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ1_HTML.gif
(1)
and its inversion formula is given by
φ ( x ) = ( h μ 1 φ ˆ ( y ) ) ( x ) = 0 ( x y ) 1 2 J μ ( x y ) φ ˆ ( y ) d y , y R + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ2_HTML.gif
(2)

where J μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq7_HTML.gif 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 φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq8_HTML.gif for μ 1 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq9_HTML.gif and 0 < θ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq10_HTML.gif as follows:
φ ˆ μ θ ( y ) = ( h μ θ φ ) ( y ) = 0 K μ θ ( x , y ) φ ( x ) d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ3_HTML.gif
(3)
where the kernel
K μ θ ( x , y ) = { c μ θ e i 2 ( x 2 + y 2 ) cot θ ( x y csc θ ) 1 2 J μ ( x y csc θ ) , θ n π , ( x y ) 1 2 J μ ( x y ) , θ = π 2 , δ ( x y ) , θ = n π , n Z , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equb_HTML.gif
and
c μ θ = exp [ i ( 1 + μ ) ( π / 2 θ ) ] sin θ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equc_HTML.gif
The inversion formula of (3) is given by
φ ( x ) = ( ( h μ θ ) 1 φ ˆ ) ( x ) = 0 K μ θ ( x , y ) ¯ ( h μ θ φ ) ( y ) d y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ4_HTML.gif
(4)
where
K μ θ ( x , y ) ¯ = exp [ i ( 1 + μ ) ( π / 2 θ ) ] e i 2 ( x 2 + y 2 ) cot θ ( x y csc θ ) 1 2 J μ ( x y csc θ ) = ( c μ θ ) ¯ sin θ e i 2 ( x 2 + y 2 ) cot θ ( x y csc θ ) 1 2 J μ ( x y csc θ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equd_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Eque_HTML.gif

We assume that throughout this paper θ n π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq11_HTML.gif, n Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq12_HTML.gif.

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
ψ b , a ( x ) = 1 a ψ ( x b a ) , b R , a > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equf_HTML.gif
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
ψ b , a , θ ( x ) = 1 a ψ ( x b a ) e i 2 ( x 2 b 2 ) cot θ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equg_HTML.gif

for all a, b and θ as above.

As per [2, 12], we defined the fractional Hankel convolution of functions φ , ψ L 1 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq13_HTML.gif as follows:
( φ # θ ψ ) ( x ) = 0 φ θ ( x , y ) ψ ( y ) d y = 0 ( τ x θ φ ) ( y ) ψ ( y ) d y = 0 φ ( y ) ( τ x θ ψ ) ( y ) d y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ5_HTML.gif
(5)
where the fractional Hankel translation of the function φ L 1 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq5_HTML.gif is defined by
( τ x θ φ ) ( y ) = φ θ ( x , y ) = e i 2 ( x 2 + y 2 ) cot θ 0 φ ( z ) D μ θ ( x , y , z ) d z , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ6_HTML.gif
(6)
and
D μ θ ( x , y , z ) = c μ θ e i 2 ( x 2 + y 2 + z 2 ) cot θ 0 ξ ( μ 1 2 ) ( x ξ csc θ ) 1 2 J μ ( x ξ csc θ ) × ( y ξ csc θ ) 1 2 J μ ( y ξ csc θ ) ( z ξ csc θ ) 1 2 J μ ( z ξ csc θ ) d ξ = 2 μ 1 2 μ 1 c μ θ e i 2 ( x 2 + y 2 + z 2 ) cot θ ( x y z ) μ 1 / 2 π Γ ( μ + 1 / 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ7_HTML.gif
(7)

where ( x , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq14_HTML.gif denotes the area of a triangle with sides x, y, z of such a triangle exists and zero otherwise. Clearly, | D μ θ ( x , y , z ) | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq15_HTML.gif and is symmetric in x, y, z.

Now, setting ξ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq16_HTML.gif, we have
0 | D μ θ ( x , y , z ) | z μ + 1 / 2 d z ( x y ) μ + 1 / 2 2 μ Γ ( μ + 1 ) | ( sin θ ) μ + 1 / 2 | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ8_HTML.gif
(8)

for x , y , z R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq17_HTML.gif.

Applying the inverse fractional Hankel transformation of D μ θ ( x , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq18_HTML.gif, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equh_HTML.gif
Lemma 1.1 If φ L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq19_HTML.gif, then
x μ 1 / 2 ( τ x θ φ ) ( y ) L 2 1 | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) φ L 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equi_HTML.gif

Proof

Since
( τ x θ φ ) ( y ) = φ θ ( x , y ) = e i 2 ( x 2 + y 2 ) cot θ 0 φ ( z ) D μ θ ( x , y , z ) d z , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equj_HTML.gif
using (8), we have
| ( τ x θ φ ) ( y ) | 0 | φ ( z ) z 1 / 2 ( μ + 1 / 2 ) { D μ θ ( x , y , z ) } 1 / 2 z 1 / 2 ( μ + 1 / 2 ) { D μ θ ( x , y , z ) } 1 / 2 | d z ( 0 z ( μ + 1 / 2 ) | φ ( z ) | 2 | D μ θ ( x , y , z ) | d z ) 1 2 ( 0 z ( μ + 1 / 2 ) | D μ θ ( x , y , z ) | d z ) 1 2 ( ( x y ) μ + 1 / 2 | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) ) 1 2 ( 0 z ( μ + 1 / 2 ) | φ ( z ) | 2 | D μ θ ( x , y , z ) | d z ) 1 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ9_HTML.gif
(9)
so that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equk_HTML.gif
Thus
x μ 1 / 2 ( τ x θ φ ) ( y ) L 2 1 | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) φ L 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equl_HTML.gif

 □

Remark 1.1 If φ L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq19_HTML.gif, then
0 | ( τ y θ φ ) ( x ) | 2 d x y 2 ( μ + 1 / 2 ) ( | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) ) 2 0 | φ ( z ) | 2 d z , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equm_HTML.gif
and
y μ 1 / 2 ( τ y θ φ ) ( x ) L 2 1 | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) φ L 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equn_HTML.gif

2 Properties of a fractional Hankel transformation

Zemanian [[5], p.129] introduced a function space H μ ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq20_HTML.gif consisting of all complex-valued infinitely differentiable function φ defined on R + = ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq6_HTML.gif, satisfying
Γ m , k μ ( φ ) = sup x R + | x m ( x 1 D ) k [ x μ 1 / 2 φ ( x ) ] | < , μ R , m , k N 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ10_HTML.gif
(10)

Definition 2.1 (Test function space H μ , θ ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq21_HTML.gif)

The space H μ , θ ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq21_HTML.gif is defined as follows: φ is a member of H μ , θ ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq21_HTML.gif if and only if it is a complex-valued C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq22_HTML.gif-function on R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq23_HTML.gif and for every choice of m and k of non-negative integers, it satisfies
ϒ m , k θ ( φ ) = sup x R + | x m Δ μ , x k φ ( x ) | < , θ n π , n Z , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ11_HTML.gif
(11)
where
Δ μ , x = [ d 2 d x 2 + 2 i x cot θ d d x + ( 1 4 μ 2 4 x 2 ) + i cot θ x 2 cot 2 θ ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ12_HTML.gif
(12)
and
Δ μ , x k = x 2 k r = 0 2 k ( l = 0 2 k a l x 2 l ) ( x 1 d d x ) r , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equo_HTML.gif

where the constants a l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq24_HTML.gif depend only on μ and parameter θ. On H μ , θ ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq21_HTML.gif, we consider the topology generated by the family { ϒ m , k θ } m , k N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq25_HTML.gif of seminorms.

Proposition 2.1 Let K μ θ ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq26_HTML.gif be the kernel of the fractional Hankel transformation. Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equp_HTML.gif

where Δ μ , x = [ d 2 d x 2 2 i x cot θ d d x + ( 1 4 μ 2 4 x 2 ) i cot θ x 2 cot 2 θ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq27_HTML.gif and is known as a fractional Bessel operator with parameter θ.

Proof See [13]. □

Example 2.1 h μ θ [ ( Δ μ , x ) r δ ( x c ) ] ( y ) = ( y 2 csc 2 θ ) r K μ θ ( c , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq28_HTML.gif, x , c R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq29_HTML.gif.

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

Proposition 2.2 Let φ L 1 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq30_HTML.gif. Then φ ˆ μ θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq31_HTML.gif satisfies the following:
(i) φ ˆ μ θ L ( R + ) with φ ˆ μ θ L A μ , θ φ L 1 , (ii) φ ˆ μ θ ( y ) 0 as y + or , (iii) φ ˆ μ θ is continuous on R + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equq_HTML.gif

where A μ , θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq32_HTML.gif is a positive constant depending on μ and θ.

Proof (i) Clearly, φ ˆ μ θ ( y ) = 0 K μ θ ( x , y ) φ ( x ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq33_HTML.gif.

So, φ ˆ μ θ L A μ , θ φ L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq34_HTML.gif.
  1. (ii)
    From Proposition 2.1(ii), where r = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq35_HTML.gif, we have
    | φ ˆ μ θ ( y ) | = 1 | ( y 2 csc 2 θ ) | | ( h μ θ ( Δ μ , x φ ( x ) ) ) ( y ) | 0 as  y ± , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equr_HTML.gif
     
although φ ˆ μ θ ( y ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq36_HTML.gif as y ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq37_HTML.gif for every φ L 1 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq30_HTML.gif.
  1. (iii)
    Let h > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq38_HTML.gif, consider
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equs_HTML.gif
     
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equt_HTML.gif
So,
sup y | φ ˆ μ θ ( y + h ) ϕ ˆ μ θ ( y ) | 0 as  h 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equu_HTML.gif

This proves that φ ˆ μ θ ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq39_HTML.gif is continuous in R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq23_HTML.gif. □

Proposition 2.3 (Parseval’s relation)

If Φ ( y ) = ( h μ θ φ ) ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq40_HTML.gif and Ψ ( y ) = ( h μ θ ψ ) ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq41_HTML.gif denote the fractional Hankel transformations of φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq8_HTML.gif and ψ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq42_HTML.gif respectively, then
0 φ ( x ) ψ ( x ) ¯ d x = sin θ 0 ( h μ θ φ ) ( y ) ( h μ θ ψ ) ( y ) ¯ d y = sin θ 0 Φ ( y ) Ψ ( y ) ¯ d y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ13_HTML.gif
(13)
and
0 | φ ( x ) | 2 d x = sin θ 0 | ( h μ θ φ ) ( y ) | 2 d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ14_HTML.gif
(14)

Proof

We have
φ , ψ = 0 φ ( x ) ψ ( x ) ¯ d x = 0 φ ( x ) ( ( c μ θ ) ¯ sin θ 0 e i 2 ( x 2 + y 2 ) cot θ ( x y csc θ ) 1 2 J μ ( x y csc θ ) ( h μ θ ψ ) ( y ) d y ) ¯ d x = c μ θ sin θ 0 ( h μ θ ψ ) ¯ ( y ) ( 0 e i 2 ( x 2 + y 2 ) cot θ ( x y csc θ ) 1 2 J μ ( x y csc θ ) φ ( x ) d x ) d y = sin θ 0 ( h μ θ φ ) ( y ) ( h μ θ ψ ) ¯ ( y ) d y = sin θ 0 Φ ( y ) Ψ ( y ) ¯ d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equv_HTML.gif
If φ = ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq43_HTML.gif, then
0 | φ ( x ) | 2 d x = sin θ 0 | ( h μ θ φ ) ( y ) | 2 d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equw_HTML.gif

 □

3 Applications of the fractional Hankel transformation to generalized differential equations

We consider the generalized n th order linear nonhomogeneous ordinary differential equation
L φ ( x ) = f ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ15_HTML.gif
(15)
where L is the generalized n th order differential operator given by
L = a n ( Δ μ , x ) n + a n 1 ( Δ μ , x ) n 1 + + a 1 ( Δ μ , x ) + a 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equx_HTML.gif

where a n , a n 1 , , a 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq44_HTML.gif are constants and Δ μ , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq45_HTML.gif is as given in Proposition 2.1.

Applying FrHT to both sides of equation (15), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equy_HTML.gif
and equivalently,
P ( y 2 csc 2 θ ) ( h μ θ φ ) ( y ) = ( h μ θ f ) ( y ) , where  P ( z ) = r = 0 n a r z r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equz_HTML.gif
Therefore,
( h μ θ φ ) ( y ) = ( h μ θ f ) ( y ) P ( y 2 csc 2 θ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ16_HTML.gif
(16)
Now, an application of the inverse FrHT gives the solution
φ ( x ) = ( h μ θ ) 1 [ ( h μ θ f ) ( y ) P ( y 2 csc 2 θ ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equaa_HTML.gif
Example 3.1 Let us consider ( 1 ( Δ μ , x ) 2 ) φ ( x ) = f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq46_HTML.gif. Then we have
φ ( x ) = ( h μ θ ) 1 [ ( 1 y 4 csc 4 θ ) 1 ( h μ θ f ) ( y ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equab_HTML.gif
Example 3.2 Using the FrHT, we investigate the solution of the generalized differential equation
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ17_HTML.gif
(17)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ18_HTML.gif
(18)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ19_HTML.gif
(19)
Let u ˆ 0 θ ( y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq47_HTML.gif be the FrHT of order zero of u ( x , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq48_HTML.gif with respect to the variable x. Then, by definition,
u ˆ 0 θ ( y , z ) = 0 K 0 θ ( x , y ) u ( x , z ) d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ20_HTML.gif
(20)

where K 0 θ ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq49_HTML.gif is the kernel of FrHT of order zero.

Taking the FrHT of order zero of (17), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equac_HTML.gif
where D d d z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq50_HTML.gif, whose solution is
u ˆ 0 θ ( y , z ) = A e z y csc θ + B e z y csc θ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ21_HTML.gif
(21)
Taking the FrHT of order zero of (18), we have
u ˆ 0 θ ( y , z ) 0 as  z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ22_HTML.gif
(22)

Condition (22) is satisfied if we have A = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq51_HTML.gif.

Therefore, from (21)
u ˆ 0 θ ( y , z ) = B e z y csc θ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ23_HTML.gif
(23)
Taking the FrHT of order zero of (19), we have
0 u ( x , 0 ) K 0 θ ( x , y ) d x = 0 f ( x ) K 0 θ ( x , y ) d x , u ˆ 0 θ ( y , 0 ) = f ˆ 0 θ ( y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ24_HTML.gif
(24)

where f ˆ 0 θ ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq52_HTML.gif is the FrHT of zero order of f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq53_HTML.gif.

Putting z = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq54_HTML.gif in (23) and using (24), we get B = f ˆ 0 θ ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq55_HTML.gif.

Hence (23) reduces to
u ˆ 0 θ ( y , z ) = f ˆ 0 θ ( y ) e z y csc θ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equad_HTML.gif
Applying the inversion formula, we have
u ( x , z ) = 0 f ˆ 0 θ ( y ) e z y csc θ K 0 θ ¯ ( x , y ) d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equae_HTML.gif

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 θ = π 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq56_HTML.gif. 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 ψ L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq57_HTML.gif which satisfies the condition
C μ , ψ , θ = 0 x 2 μ 2 | ( h μ θ ψ ) ( x ) | 2 d x < , μ 1 / 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equaf_HTML.gif
where C μ , ψ , θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq58_HTML.gif is called the admissibility condition of the fractional Bessel wavelet and ( h μ θ ψ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq59_HTML.gif is the fractional Hankel transformation of ψ. The fractional Bessel wavelets ψ b , a θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq60_HTML.gif are generated from one single function ψ L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq57_HTML.gif by dilation and translation with parameters a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq61_HTML.gif and b 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq62_HTML.gif respectively by
ψ b , a θ ( x ) = 1 a D a τ b θ ψ ( x ) = 1 a D a ψ θ ( b , x ) = 1 a ψ θ ( b a , x a ) = 1 a e i 2 ( b 2 a 2 + x 2 a 2 ) cot θ 0 ψ ( z ) D μ θ ( b a , x a , z ) d z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equag_HTML.gif
Lemma 4.1 If ψ L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq63_HTML.gif, then
ψ b , a θ L 2 b ( μ + 1 / 2 ) a ( μ + 1 / 2 ) | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) ψ L 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equah_HTML.gif

Proof

We have
ψ b , a θ ( x ) = 1 a e i 2 ( b 2 a 2 + x 2 a 2 ) cot θ 0 ψ ( z ) D μ θ ( b a , x a , z ) d z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equai_HTML.gif
Now,
| ψ b , a θ ( x ) | 1 a 0 | ψ ( z ) | z 1 / 2 ( μ + 1 / 2 ) | { D μ θ ( b a , x a , z ) } 1 / 2 | z 1 / 2 ( μ + 1 / 2 ) × | { D μ θ ( b a , x a , z ) } 1 / 2 | d z 1 a ( 0 z ( μ + 1 / 2 ) | ψ ( z ) | 2 | D μ θ ( b a , x a , z ) | d z ) 1 / 2 × ( 0 | D μ θ ( b a , x a , z ) | z ( μ + 1 / 2 ) d z ) 1 / 2 1 a ( ( b x ) μ + 1 / 2 a 2 ( μ + 1 / 2 ) | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) ) 1 / 2 × ( 0 z ( μ + 1 / 2 ) | ψ ( z ) | 2 | D μ θ ( b a , x a , z ) | d z ) 1 / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equaj_HTML.gif
Therefore,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equak_HTML.gif
Thus,
ψ b , a θ L 2 b ( μ + 1 / 2 ) a ( μ + 1 / 2 ) | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) ψ L 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equal_HTML.gif

 □

Theorem 4.1 Let f , ψ L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq64_HTML.gif. Then the continuous fractional Bessel wavelet transformation B ψ θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq65_HTML.gif is defined on f by
( B ψ θ f ) ( b , a ) = a μ sin θ c μ θ ¯ 0 e i 2 ( 1 a 2 1 ) ( a x ) 2 cot θ x μ 1 2 ( b x csc θ ) 1 2 × J μ ( b x csc θ ) ( h μ θ e i 2 ( ) 2 cot θ f ) ( x ) ( h μ θ ψ ) ¯ ( a x ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equam_HTML.gif

Proof

We have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equan_HTML.gif
by putting ξ a = x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq66_HTML.gif, then the continuous fractional Bessel wavelet transformation can be written as
( B ψ θ f ) ( b , a ) = a μ sin θ c μ θ ¯ 0 e i 2 ( 1 a 2 1 ) ( a x ) 2 cot θ x μ 1 2 ( b x csc θ ) 1 2 × J μ ( b x csc θ ) ( h μ θ e i 2 ( ) 2 cot θ f ) ( x ) ( h μ θ ψ ) ¯ ( a x ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equao_HTML.gif
This means that
h μ θ { e i 2 b 2 cot θ ( B ψ θ f ) ( b , a ) } = a μ sin θ ( x μ 1 2 e i 2 a 2 x 2 cot θ ( h μ θ e i 2 ( ) 2 cot θ f ) ( x ) ( h μ θ ψ ) ¯ ( a x ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equap_HTML.gif

 □

Remark 4.1 If f L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq67_HTML.gif is a homogeneous function of degree n, then
( B ψ θ f ) ( λ b , λ a ) = λ n + 1 2 ( B ψ θ f ) ( b , a ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equaq_HTML.gif
Theorem 4.2 If ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq68_HTML.gif and ψ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq69_HTML.gif are two wavelets and ( B ψ 1 θ f ) ( b , a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq70_HTML.gif and ( B ψ 2 θ g ) ( b , a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq71_HTML.gif denote the continuous fractional Bessel wavelet transformations of f , g L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq72_HTML.gif respectively, then
0 0 ( B ψ 1 θ f ) ( b , a ) ( B ψ 2 θ g ) ¯ ( b , a ) d b d a a 2 = sin 2 θ C μ , ψ 1 , ψ 2 , θ f , g , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equar_HTML.gif
where
C μ , ψ 1 , ψ 2 , θ = 0 a 2 μ 2 ( h μ θ ψ 1 ) ¯ ( a ) ( h μ θ ψ 2 ) ( a ) d a < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equas_HTML.gif

Proof

We have
( B ψ 1 θ f ) ( b , a ) = a μ sin θ c μ θ ¯ 0 e i 2 ( 1 a 2 1 ) ( a x ) 2 cot θ x μ 1 2 ( b x csc θ ) 1 2 × J μ ( b x csc θ ) ( h μ θ e i 2 ( ) 2 cot θ f ) ( x ) ( h μ θ ψ 1 ) ¯ ( a x ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equat_HTML.gif
Now,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equau_HTML.gif

 □

Theorem 4.3 If ψ is a wavelet and ( B ψ θ f ) ( b , a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq73_HTML.gif and ( B ψ θ g ) ( b , a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq74_HTML.gif are the continuous fractional Bessel wavelet transformations of f , g L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq75_HTML.gif respectively, then
0 0 ( B ψ θ f ) ( b , a ) ( B ψ θ g ) ¯ ( b , a ) d b d a a 2 = sin 2 θ C μ , ψ , θ f , g . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equav_HTML.gif

Proof The proof of Theorem 4.3 can be easily deduced by setting ψ 1 = ψ 2 = ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq76_HTML.gif in Theorem 4.2. □

Remark 4.2 If f = g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq77_HTML.gif and ψ 1 = ψ 2 = ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq76_HTML.gif, then from Theorem 4.3, we have
0 0 | ( B ψ θ f ) ( b , a ) | 2 d b d a a 2 = sin 2 θ C μ , ψ , θ f 2 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equaw_HTML.gif
Theorem 4.4 Let f L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq67_HTML.gif. Then f can be reconstructed by the formula
f ( t ) = 1 sin 2 θ C μ , ψ , θ 0 0 ( B ψ θ f ) ( b , a ) ψ b , a θ ( t ) d b d a a 2 , a > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equax_HTML.gif
Proof For any g L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq78_HTML.gif, we have
sin 2 θ C μ , ψ , θ f , g = 0 0 ( B ψ θ f ) ( b , a ) ( B ψ θ g ) ¯ ( b , a ) d b d a a 2 = 0 0 ( B ψ θ f ) ( b , a ) ( 0 g ( t ) ψ b , a θ ( t ) ¯ d t ¯ ) d b d a a 2 = 0 [ 0 0 ( B ψ θ f ) ( b , a ) ψ b , a θ ( t ) d b d a a 2 ] g ( t ) ¯ d t = 0 0 ( B ψ θ f ) ( b , a ) ψ b , a θ ( t ) d b d a a 2 , g ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equay_HTML.gif
Therefore,
f ( t ) = 1 sin 2 θ C μ , ψ , θ 0 0 ( B ψ θ f ) ( b , a ) ψ b , a θ ( t ) d b d a a 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equaz_HTML.gif

 □

Theorem 4.5 If ψ L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq57_HTML.gif, then
0 [ ( B ψ θ f ) ( b , a ) ( B ψ θ g ) ( b , a ) ¯ ] d b = a 2 μ sin 2 θ F , G , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equba_HTML.gif
where
{ F ( x ) : = e i 2 ( ( 2 a 2 ) x 2 ) cot θ x μ 1 2 ( h μ θ e i 2 ( ) 2 cot θ f ) ( x ) ( h μ θ ψ ) ¯ ( a x ) , G ( x ) : = e i 2 ( ( 2 a 2 ) x 2 ) cot θ x μ 1 2 ( h μ θ e i 2 ( ) 2 cot θ g ¯ ) ( x ) ( h μ θ ψ ) ¯ ( a x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbb_HTML.gif

Proof

Using Theorem 4.1 and Theorem 4.2, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbc_HTML.gif

This completes the proof of the theorem. □

Theorem 4.6 If ψ L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq63_HTML.gif is a Bessel wavelet and f is a bounded integrable function, then the convolution ( ψ θ f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq79_HTML.gif is a fractional Bessel wavelet, where
( ψ θ f ) ( x ) = 0 ( τ x θ ψ ) ( y ) y ( μ + 1 / 2 ) f ( y ) d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbd_HTML.gif

Proof

We have
( ψ θ f ) ( x ) = 0 ( τ x θ ψ ) ( y ) y ( μ + 1 / 2 ) f ( y ) d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Eqube_HTML.gif
Therefore,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbf_HTML.gif
This implies that
( ψ θ f ) ( x ) L 2 1 | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) ψ L 2 f L 1 < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbg_HTML.gif
We have ( ψ θ f ) L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq80_HTML.gif. Moreover,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbh_HTML.gif

Thus, the convolution function ( ψ θ f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq79_HTML.gif is a fractional Bessel wavelet. □

Theorem 4.7 If f , ψ L 2 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq81_HTML.gif and ( B ψ θ f ) ( b , a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq82_HTML.gif is the continuous fractional Bessel wavelet transformation, then
(i) ( B ψ θ f ) ( b , a ) is continuous on R + × R + , (ii) ( B ψ θ f ) ( b , a ) L b ( μ + 1 / 2 ) a ( μ + 1 / 2 ) | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) f L 2 ψ L 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbi_HTML.gif
Proof (i) Let ( b 0 , a 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq83_HTML.gif be an arbitrary but fixed point in R + × R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq84_HTML.gif. Then, by the Hölder inequality,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbj_HTML.gif
Since
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbk_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbl_HTML.gif
by the dominated convergence theorem and the continuity of D μ θ ( b a , x a , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq85_HTML.gif in the variable b and a, we have
lim b b 0 lim a a 0 | ( B ψ θ f ) ( b , a ) ( B ψ θ f ) ( b 0 , a 0 ) | = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbm_HTML.gif
This proves that ( B ψ θ f ) ( b , a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq82_HTML.gif is continuous on R + × R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq84_HTML.gif.
  1. (ii)
    We have
    ( B ψ θ f ) ( b , a ) = 0 f ( x ) ( 1 a e i 2 ( b 2 a 2 + x 2 a 2 ) cot θ 0 ψ ( z ) D μ θ ( b a , x a , z ) d z ) ¯ d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbn_HTML.gif
     
Therefore, by the Hölder inequality, we have
| ( B ψ θ f ) ( b , a ) | 1 a ( 0 0 x ( μ + 1 / 2 ) | f ( x ) | 2 z ( μ + 1 / 2 ) | D μ θ ( b a , x a , z ) | d x d z ) 1 / 2 × ( 0 0 z ( μ + 1 / 2 ) | ψ ( z ) | 2 x ( μ + 1 / 2 ) | D μ θ ( b a , x a , z ) | d x d z ) 1 / 2 = 1 a ( 0 x ( μ + 1 / 2 ) | f ( x ) | 2 d x 0 | D μ θ ( b a , x a , z ) | z ( μ + 1 / 2 ) d z ) 1 / 2 × ( 0 z ( μ + 1 / 2 ) | ψ ( z ) | 2 d z 0 | D μ θ ( b a , x a , z ) | x ( μ + 1 / 2 ) d x ) 1 / 2 1 a ( b μ + 1 / 2 a 2 ( μ + 1 / 2 ) | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) ) 1 / 2 ( 0 | f ( x ) | 2 d x ) 1 / 2 × ( b μ + 1 / 2 a 1 | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) ) 1 / 2 ( 0 | ψ ( z ) | 2 d z ) 1 / 2 = b ( μ + 1 / 2 ) a ( μ + 1 / 2 ) | ( sin θ ) μ + 1 / 2 | 2 μ Γ ( μ + 1 ) f L 2 ψ L 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbo_HTML.gif

 □

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

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

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