The continuous fractional Bessel wavelet transformation

  • Akhilesh Prasad1,

    Affiliated with

    • Ashutosh Mahato1,

      Affiliated with

      • Vishal Kumar Singh1 and

        Affiliated with

        • Madan Mohan Dixit2Email author

          Affiliated with

          Boundary Value Problems20132013:40

          DOI: 10.1186/1687-2770-2013-40

          Received: 17 November 2012

          Accepted: 25 January 2013

          Published: 27 February 2013

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          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 ) http://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 ℝ such that the integral R | ϕ ( x ) | p d t http://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 ) http://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 = . http://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 http://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 + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq5_HTML.gif, R + = ( 0 , ) http://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 , http://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 + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ2_HTML.gif
          (2)

          where J μ http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq8_HTML.gif for μ 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq9_HTML.gif and 0 < θ < π http://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 , http://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 , http://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 . http://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 , http://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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equd_HTML.gif
          and
          http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq11_HTML.gif, n Z http://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 , http://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 , http://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 + ) http://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 , http://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 + ) http://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 , http://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 ) , http://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 ) http://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 http://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 http://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 | http://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 + http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq18_HTML.gif, we obtain
          http://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 + ) http://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 . http://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 , http://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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ9_HTML.gif
          (9)
          so that
          http://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 . http://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 + ) http://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 , http://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 . http://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 + ) http://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 , ) http://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 . http://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 + ) http://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 + ) http://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 + ) http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq22_HTML.gif-function on R + http://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 , http://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 ] , http://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 , http://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 http://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 + ) http://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 http://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 ) http://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
          http://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 ] http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq28_HTML.gif, x , c R + http://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 + ) http://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 http://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 + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equq_HTML.gif

          where A μ , θ http://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 http://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 http://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 http://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 ± , http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq36_HTML.gif as y ± http://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 + ) http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq38_HTML.gif, consider
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equs_HTML.gif
             
          and
          http://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 . http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq39_HTML.gif is continuous in R + http://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 ) http://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 ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq8_HTML.gif and ψ ( x ) http://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 , http://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 . http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equv_HTML.gif
          If φ = ψ http://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 . http://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 ) , http://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 , http://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 http://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 http://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
          http://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 . http://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 ) . http://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 ) ] . http://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 ) http://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 ) ] . http://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
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ17_HTML.gif
          (17)
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equ18_HTML.gif
          (18)
          http://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 ) http://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 ) http://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 , http://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 ) http://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
          http://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 http://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 . http://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 . http://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 http://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 . http://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 ) , http://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 ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq53_HTML.gif.

          Putting z = 0 http://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 ) http://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 . http://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 . http://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 http://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 + ) http://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 , http://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 http://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 ) http://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 http://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 + ) http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq61_HTML.gif and b 0 http://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 . http://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 + ) http://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 . http://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 . http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equaj_HTML.gif
          Therefore,
          http://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 . http://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 + ) http://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 http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equam_HTML.gif

          Proof

          We have
          http://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 http://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 . http://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 ) ) . http://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 + ) http://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 ) . http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq68_HTML.gif and ψ 2 http://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 ) http://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 ) http://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 + ) http://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 , http://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 < . http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equat_HTML.gif
          Now,
          http://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 ) http://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 ) http://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 + ) http://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 . http://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 http://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 http://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 http://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 . http://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 + ) http://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 . http://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 + ) http://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 ) . http://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 . http://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 + ) http://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 , http://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 ) . http://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
          http://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 + ) http://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 ) http://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 . http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Eqube_HTML.gif
          Therefore,
          http://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 < . http://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 + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_IEq80_HTML.gif. Moreover,
          http://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 ) http://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 + ) http://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 ) http://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 . http://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 ) http://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 + http://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,
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbj_HTML.gif
          Since
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-40/MediaObjects/13661_2012_Article_294_Equbk_HTML.gif
          and
          http://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 ) http://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 . http://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 ) http://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 + http://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 . http://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 . http://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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