A General Geometric Fourier Transform Convolution Theorem

TL;DR

This paper extends the general geometric Fourier transform framework by establishing a convolution theorem, broadening its theoretical foundation and applicability across various geometric algebra Fourier transforms.

Contribution

It introduces a convolution theorem for the general geometric Fourier transform, expanding the theoretical understanding and potential applications of these transforms.

Findings

Established a convolution theorem for the general geometric Fourier transform.

Identified necessary constraints for properties like linearity, scaling, and shifting.

Extended previous results to include convolution, enhancing the transform's theoretical framework.

Abstract

The large variety of Fourier transforms in geometric algebras inspired the straight forward definition of ``A General Geometric Fourier Transform`` in Bujack et al., Proc. of ICCA9, covering most versions in the literature. We showed which constraints are additionally necessary to obtain certain features like linearity, a scaling, or a shift theorem. In this paper we extend the former results by a convolution theorem.