Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl_3,0

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 Cl_3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl_3,0 multivector functions.