IJPAM: Volume 55, No. 2 (2009)

THE HANKEL CONVOLUTION OF ARBITRARY ORDER

C.K. Li
Department of Mathematics and Computer Science
Brandon University
Brandon, Manitoba, R7A 6A9, CANADA
e-mail: [email protected]


Abstract.Let $\mu > -1/2$. The classical Hankel transform is defined by

\begin{displaymath}
(h_{\mu}\phi) (t) = \int_{0}^{\infty} x J_{\mu}(xt) \phi(x) d x
\quad t \in (0, \infty),
\end{displaymath} (1)

where $J_{\mu}(x)$ denotes the Bessel function of the first kind and order $\mu$. The goal of this paper is to construct the Hankel transform of arbitrary order $h_{\mu, k}$ based on the two differential operators and show that $h_{\mu, k} = h_{\mu, k}^{-1}$ on $H_{\mu - \frac{1}{2}}$ for $\mu \in R$. Further more, the Hankel convolution of arbitrary order is introduced with the following identity

\begin{displaymath}
(h_{\mu, k}h )(t) = t^{- \mu} (h_{\mu, k}\phi)(t) (h_{\mu,
k}\psi)(t)
\end{displaymath}

on the spaces $(H_{\mu - \frac{1}{2}}, \, H_{\mu - \frac{1}{2}})$ and $(S_{\mu}, \, H_{\mu - \frac{1}{2}})$ respectively.

Received: August 1, 2009

AMS Subject Classification: 46F12

Key Words and Phrases: Hankel transform, Bessel function, Hankel convolution and automorphism

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 55
Issue: 2