Generalized translation operator and the Heat equation for the canonical Fourier Bessel transform

TL;DR

This paper introduces a new translation operator for the Fourier Bessel transform, explores its properties, and applies it to study the heat equation and heat semigroup related to a specific differential operator.

Contribution

It defines a generalized translation operator for the Fourier Bessel transform and investigates its properties and applications to the heat equation.

Findings

Established a convolution product for the Fourier Bessel transform.

Derived properties of the translation operator.

Analyzed the heat equation and heat semigroup associated with the operator.

Abstract

The aim of this paper is to introduce a translation operator associated to the canonical Fourier Bessel transform $\mathcal{F}_{\nu}^{\mathbf{m}}$ and study some of the important properties. We derive a convolution product for this transform and as application we study the heat equation and the heat semigroup related to $\Delta_{\nu}^{\mathbf{m}}.$