OPS-QFTs: A new type of quaternion Fourier transforms based on the orthogonal planes split with one or two general pure quaternions

TL;DR

This paper introduces a novel quaternion Fourier transform based on the orthogonal planes split (OPS) using one or two pure quaternions, providing a geometric interpretation and inverse transformations.

Contribution

It generalizes quaternionic Fourier transforms by incorporating the OPS with arbitrary pure quaternions, enhancing geometric understanding and mathematical flexibility.

Findings

Defined OPS based on pure quaternions f,g

Generalized quaternion Fourier transform using OPS

Established inverse transformations for the new QFT

Abstract

We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions $f,g$. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by $f,g$, or by only one pure unit quaternion $f$, comment on their geometric meaning, and establish inverse transformations. Keywords: Clifford geometric algebra, quaternion geometry, quaternion Fourier transform, inverse Fourier transform, orthogonal planes split