A General Geometric Fourier Transform

TL;DR

This paper introduces a unified definition of the geometric Fourier transform that encompasses most existing versions, providing a comprehensive framework with standard theorems applicable across different variants.

Contribution

It offers a general, straightforward definition of the geometric Fourier transform and establishes necessary constraints for specific features, streamlining future development.

Findings

Provides a unified framework for geometric Fourier transforms

Establishes conditions for linearity and shift theorems

Standard theorems are proved once for all variants

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.