The class of Clifford-Fourier transforms

TL;DR

This paper explores the Clifford-Fourier transform, identifying a broad class of solutions to its defining PDEs, deriving explicit series expressions, and analyzing properties and inverses in even dimensions.

Contribution

It introduces an entire class of solutions to the Clifford-Fourier PDE system, expanding understanding beyond the classical unique solution.

Findings

Derived series expressions using Gegenbauer polynomials and Bessel functions.

Computed eigenvalues for the associated integral transforms.

Established inverse transforms in even-dimensional cases.

Abstract

Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called Clifford-Fourier transform (see [F. Brackx et al., The Clifford-Fourier transform. J. Fourier Anal. Appl. 11 (2005), 669--681]). The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the unique solution. Here we determine an entire class of solutions of this system of PDEs, under certain constraints. For each solution, series expressions in terms of Gegenbauer polynomials and Bessel functions are obtained. This allows to compute explicitly the eigenvalues of the associated integral transforms. In the even-dimensional case, this also yields the inverse transform for each of the solutions. Finally, several properties of the entire class of solutions are proven.