Fractional Jacobi-Dunkl transform: properties and application

Abstract

The aim of this paper is to develop a novel harmonic analysis in the Jacobi-Dunkl setting. We propose the fractional Jacobi-Dunkl transform in order to generalize the fractional Fourier transform. Firstly, we investigate the associated harmonic analysis and we derive some of its basic properties, such as inversion formula and Parseval identity. In sequel, we derive the fractional Jacobi-Dunkl transform on the corresponding Schwartz space and tempered distribution. Finally, we extend the scope of our work by investigating the fractional Jacobi-Dunkl transform for heat equation.