The Class of Clifford-Fourier Transforms

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.