The continuous fractional Bessel wavelet transformation
© Prasad et al.; licensee Springer. 2013
Received: 17 November 2012
Accepted: 25 January 2013
Published: 27 February 2013
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.
KeywordsHankel transformation fractional Hankel transformation fractional Bessel wavelet transformation Bessel function
Pathak and Dixit  introduced continuous and discrete Bessel wavelet transformations and studied their properties by exploiting the Hankel convolution of Haimo  and Hirschman . Upadhyay et al.  studied the continuous Bessel wavelet transformation associated with the Hankel-Hausdorff operator.
where is the Bessel function of the first kind of order μ.
We assume that throughout this paper , .
for all a, b and θ as above.
where denotes the area of a triangle with sides x, y, z of such a triangle exists and zero otherwise. Clearly, and is symmetric in x, y, z.
2 Properties of a fractional Hankel transformation
Definition 2.1 (Test function space )
where the constants depend only on μ and parameter θ. On , we consider the topology generated by the family of seminorms.
where and is known as a fractional Bessel operator with parameter θ.
Proof See . □
Example 2.1 , .
The result can be easily shown by using Proposition 2.1(ii).
where is a positive constant depending on μ and θ.
Proof (i) Clearly, .
- (ii)From Proposition 2.1(ii), where , we have
- (iii)Let , consider
This proves that is continuous in . □
Proposition 2.3 (Parseval’s relation)
3 Applications of the fractional Hankel transformation to generalized differential equations
where are constants and is as given in Proposition 2.1.
where is the kernel of FrHT of order zero.
Condition (22) is satisfied if we have .
where is the FrHT of zero order of .
Putting in (23) and using (24), we get .
4 The continuous fractional Bessel wavelet transformation
The continuous fractional Bessel wavelet transformation (CFrBWT) is a generalization of the ordinary continuous Bessel wavelet transformation (CBWT) with parameter θ, that is, CBWT is a special case of CFrBWT with parameter . In this section, we define the continuous fractional Bessel wavelet transformation and study some of its properties using the theory of fractional Hankel convolution (5) corresponding to .
Proof The proof of Theorem 4.3 can be easily deduced by setting in Theorem 4.2. □
This completes the proof of the theorem. □
Thus, the convolution function is a fractional Bessel wavelet. □
- (ii)We have
Dedicated to Professor Hari M Srivastava.
- Pathak RS, Dixit MM: Continuous and discrete Bessel wavelet transforms. J. Comput. Appl. Math. 2003, 160(1-2):241-250. 10.1016/S0377-0427(03)00626-5MATHMathSciNetView Article
- Haimo DT: Integral equations associated with Hankel convolution. Trans. Am. Math. Soc. 1965, 116: 330-375.MATHMathSciNetView Article
- Hirschman II Jr.: Variation diminishing Hankel transform. J. Anal. Math. 1960-1961, 8: 307-336. 10.1007/BF02786854MATHMathSciNetView Article
- Upadhyay SK, Yadav RN, Debnath L: On continuous Bessel wavelet transformation associated with the Hankel-Hausdorff operator. Integral Transforms Spec. Funct. 2012, 23(5):315-323. 10.1080/10652469.2011.586156MATHMathSciNetView Article
- Zemanian AH: Generalized Integral Transformations. Interscience, New York; 1968.MATH
- Namias V: Fractionalization of Hankel transform. J. Inst. Math. Appl. 1980, 26: 187-197. 10.1093/imamat/26.2.187MATHMathSciNetView Article
- Sheppard CJR, Larkin KG: Similarity theorems for fractional Fourier transforms and fractional Hankel transforms. Opt. Commun. 1998, 154: 173-178. 10.1016/S0030-4018(98)00229-6View Article
- Zhang Y, Funaba T, Tanno N: Self-fractional Hankel functions and their properties. Opt. Commun. 2000, 176: 71-75. 10.1016/S0030-4018(00)00518-6View Article
- Kerr FH: Fractional powers of Hankel transforms in the Zemanian spaces. J. Math. Anal. Appl. 1992, 166: 65-83. 10.1016/0022-247X(92)90327-AMATHMathSciNetView Article
- Debnath L: Wavelet Transforms and Their Applications. Birkhäuser, Boston; 2002.MATHView Article
- Shi J, Zhang N, Liu X: A novel fractional wavelet transform and its applications. Sci. China Inf. Sci. 2012, 55(6):1270-1279. 10.1007/s11432-011-4320-xMATHMathSciNetView Article
- Betancor JJ, Gonzales BJ: A convolution operation for a distributional Hankel transformation. Stud. Math. 1995, 117(1):57-72.MATH
- Prasad A, Singh VK: The fractional Hankel transform of certain tempered distributions and pseudo-differential operators. Ann. Univ. Ferrara 2012. doi:10.1007/s11565-012-0169-1