Generalized Jacobi transform

Abstract

Recently, Koornwinder and Walter derived an inversion formula for the finite continuous Jacobi transform for all α, β > −1. This inversion formula generalizes the one obtained earlier by Walter and Zayed for α,β > −1 and α+ β is a non-negative integer.

In this paper we extend the finite continuous Jacobi transform and its inversion formula as obtained by Koornwinder and Walter to generalized functions. In particular, a fundamental space will be constructed and the generalized transform will be defined on the dual space. Several properties of the generalized transform will be studied along with a generalized inversion formula. Some examples of the finite continuous Jacobi transform and its inversion formula will also be given.

Keywords:

• Jacobi transform
• generalized functions
• inverse Jacobi transform
• sampling theorems

AMS:

• 44A15
• 33A65
• 33A3

:^∗On leave from California Polytechnic State University, San Luis Obispo, CA 93407

:^∗On leave from California Polytechnic State University, San Luis Obispo, CA 93407

Notes

:^∗On leave from California Polytechnic State University, San Luis Obispo, CA 93407