Clifford Fourier transform on multivector fields and uncertainty principles for dimensions n=2 (mod 4) and n=3 (mod 4)

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 : R-n -> Cl-n,Cl-0, n = 2,3 (mod 4)). Third, we show a set of important properties of the Clifford Fourier transform on Cl-n,Cl-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 fx(m), f del(m) and for the Clifford convolution. Finally, we apply Clifford Fourier transform properties for proving an uncertainty principle for Cl-n,Cl-0, n = 2,3 (mod 4) multivector functions.