A new construction of the Clifford-Fourier kernel

TL;DR

This paper introduces a novel Laplace transform-based method to derive explicit formulas and integral representations for the Clifford-Fourier kernel across all dimensions, enhancing understanding of its structure.

Contribution

It presents a new approach using Laplace transforms to explicitly compute the Clifford-Fourier kernel, including generating functions and integral representations.

Findings

Explicit kernel expressions as finite sums of Bessel functions in even dimensions

New integral representations for the Clifford-Fourier kernel in all dimensions

Derived the formal generating function for even-dimensional kernels

Abstract

In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fourier transform. First, the kernel of the Clifford-Fourier transform in the Laplace domain is obtained. When the dimension is even, the inverse Laplace transform may be computed and we obtain the explicit expression for the kernel as a finite sum of Bessel functions. We equally obtain the plane wave decomposition and find new integral representations for the kernel in all dimensions. Finally we define and compute the formal generating function for the even dimensional kernels.