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Clifford Fourier transform on multivector fields and uncertainty principles for dimensions n=2(mod4) and n=3(mod4). (English) Zbl 1177.15029
Summary: 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: n Cl n,0 , n=2,3(mod4)). Third, we show a set of important properties of the Clifford Fourier transform on Cl n,0 , n=2,3(mod4) 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 m and for the Clifford convolution. Finally, we apply Clifford Fourier transform properties for proving an uncertainty principle for Cl n,0 , n=2,3(mod4) multivector functions.
MSC:
15A66Clifford algebras, spinors
43A32Other transforms and operators of Fourier type