Linear Canonical Jacobi-Dunkl Transform: Theory and Applications

TL;DR

This paper introduces the linear canonical Jacobi-Dunkl transform (LCJDT), a novel harmonic analysis tool combining JDT and LCT, with theoretical properties and applications to differential equations.

Contribution

The paper develops the LCJDT, deriving its kernel, properties, and demonstrating its application to solving heat equations, advancing harmonic analysis methods.

Findings

Derived the kernel function of LCJDT

Established inversion and Parseval's theorem for LCJDT

Applied LCJDT to solve heat equation

Abstract

This paper aims to develop an innovative method for harmonic analysis by introducing the linear canonical Jacobi-Dunkl transform (LCJDT), which integrates both the Jacobi-Dunkl transform (JDT) and the linear canonical transform (LCT). Firstly, the kernel function of the LCJDT is derived, and its fundamental properties are examined. Subsequently, the LCJDT is established, along with an investigation of its essential properties, including the inversion formula, Parseval's theorem, differentiation, the convolution theorem, and the uncertainty principle. Finally, the potential application of the LCJDT in solving the heat equation is explored.