Directional Uncertainty Principle for Quaternion Fourier Transform

TL;DR

This paper establishes a new directional uncertainty principle for quaternion Fourier transforms, extending it to Clifford algebras and demonstrating its application to spacetime algebra functions.

Contribution

It introduces a novel directional uncertainty principle for quaternion Fourier transforms and generalizes it to Clifford geometric algebras, with practical examples.

Findings

Derived a new directional uncertainty principle for quaternion Fourier transforms.

Extended the principle to Clifford geometric algebras with quaternion subalgebras.

Applied the principle to a spacetime algebra function example.

Abstract

This paper derives a new directional uncertainty principle for quaternion valued functions subject to the quaternion Fourier transformation. This can be generalized to establish directional uncertainty principles in Clifford geometric algebras with quaternion subalgebras. We demonstrate this with the example of a directional spacetime algebra function uncertainty principle related to multivector wave packets.