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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