Uncertainty Principle for the Clifford Geometric Algebra Cl(3,0) based on Clifford Fourier Transform

TL;DR

This paper extends the Fourier transform into Clifford geometric algebra for Cl(3,0) and establishes an uncertainty principle specific to multivector functions within this algebraic framework.

Contribution

It introduces a detailed Clifford Fourier transform for Cl(3,0) and proves an uncertainty principle tailored to multivector functions in this algebra.

Findings

Defined the Clifford Fourier transform for Cl(3,0)

Proved an uncertainty principle for multivector functions

Enhanced understanding of signal analysis in geometric algebra

Abstract

In the field of applied mathematics the Fourier transform has developed into an important tool. It is a powerful method for solving partial differential equations. The Fourier transform provides also a technique for signal analysis where the signal from the original domain is transformed to the spectral or frequency domain. In the frequency domain many characteristics of the signal are revealed. With these facts in mind, we extend the Fourier transform in geometric algebra. We explicitly show detailed properties of the real Clifford geometric algebra Fourier transform (CFT), which we subsequently use to define and prove the uncertainty principle for Cl(3,0) multivector functions.