Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions n = 2 (mod 4) and n = 3 (mod 4)

Abstract

First, the basic concepts of the multivector functions, vector differential and vector derivative in geometric algebra are introduced. Second, we define a generalized real Fourier transform on Clifford multivector-valued functions (\(f : {{\mathbb{R}}}^n \rightarrow Cl_{n,0}, n = 2, 3\) (mod 4)). Third, we show a set of important properties of the Clifford Fourier transform on Cln,0, n = 2, 3 (mod 4) such as differentiation properties, and the Plancherel theorem, independent of special commutation properties. Fourth, we develop and utilize commutation properties for giving explicit formulas for fxm, f ∇m and for the Clifford convolution. Finally, we apply Clifford Fourier transform properties for proving an uncertainty principle for Cln,0, n = 2, 3 (mod 4) multivector functions.