R. Bujack, G. Scheuermann, E. Hitzer
A General Geometric Fourier Transform
In K. Guerlebeck (ed.), Electronic Proceedings of The 9th International Conference on Clifford Algebras and their Applications in Mathematical Physics (ICCA9), 15-10 July 2011, Weimar, Germany, (2011).

Abstract: The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature. We show which constraints are additionally necessary to obtain certain features like linearity or a shift theorem. As a result, we provide guidelines for the target-oriented design of yet unconsidered transforms that fulfill requirements in a specific application context. Furthermore, the standard theorems do not need to be shown in a slightly different form every time a new geometric Fourier transform is developed since they are proved here once and for all.

Keywords: Fourier transform, geometric algebra, Clifford algebra, image processing, linearity, scaling, shift.

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