Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl3,0

Abstract

First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions \( (f:\user2{\mathbb{R}}^3 \to Cl_{3,0} ). \) Third, we show a set of important properties of the Clifford Fourier transform on Cl3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl3,0 multivector functions.