Demystification of the geometric Fourier transforms and resulting convolution theorems

Abstract

Communicated by T. QianAs it will turn out in this paper, the recent hype about most of the Clifford-Fourier transforms is not thoroughly worth the pain. Almost everyone that has a real application is separable, and these transforms can be decomposed into a sum of real valued transforms with constant multivecor factors. This fact makes their interpretation, their analysis, and their implementation almost trivial.

New Applications of Clifford’s Geometric Algebra

“…The QDFT is defined and its main properties are analyzed, including quaternion dilation, modulation and shift properties, Plancherel and Parseval identities, covariance under orthogonal transformations, transformations of coordinate polynomials and differential operator polynomials, transformations of derivative and Dirac derivative operators, as well as signal width related to band width uncertainty relationships. In [26] many application relevant Clifford Fourier transforms are shown to be separable and decomposable into sums of real valued transforms with constant multivector factors. This fact eases their interpretation, their analysis, implementation and application.…”