The Class of Clifford-Fourier Transforms

Abstract

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 . 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.

Fractional Fourier transforms of hypercomplex signals

“…In the paper [13] we started our analysis from 4 different, yet equivalent, definitions of the classical Fourier transform in R m . Each of these formulations allows for generalization to hypercomplex FTs.…”

“…So far, we have applied these ideas in 3 different directions of hypercomplex FTs, namely -k-vector Fourier transforms ( [14]) -radially deformed Fourier transforms ( [15,16]) -Clifford-Fourier transforms ( [17,13]). …”

“…In recent work (see [13][14][15][16][17]) we have developed a different methodology: we start from a list of properties or general mathematical principles we want a hypercomplex Fourier transform to have, and then determine all kernels that satisfy these properties.…”

“…In particular we devote our attention to the so-called Clifford-Fourier transform (as studied in [17,13]). Subsequently, we introduce a new fractional version of the Clifford-Fourier transform (CFT).…”

Quaternion and Clifford Fourier Transforms and Wavelets

“…(v) the Clifford-Fourier transform and the fractional Clifford-Fourier transform, both already mentioned above; meanwhile an entire class of CliffordFourier transforms has been thoroughly studied in [36]; (vi) the radially deformed hypercomplex Fourier transform, which appears as a special case in the theory of radial deformations of the Lie algebra osp(1|2), see [38,37], and is a topic of current research, see [32].…”