E.M.S. Hitzer, B. Mawardi
Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions n = 2 (mod 4) and n = 3 (mod 4)
P. Angles (ed.), Adv. App. Cliff. Alg., Vol. 18, S3,4, pp. 715-736 (2008).

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 : R^n --> Cl(n,0), n = 2, 3 (mod 4) ). Third, we show a set of important properties of the Clifford Fourier transform on Cl(n,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 f x^m , f Delta^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 (mod 4) multivector functions.

(*) f x^m ... moments with vector variable x, f Delta^m ... m-fold vector derivatives Delta.

Keywords. Vector derivative, multivector-valued function, Clifford (geometric) algebra, Clifford Fourier transform, uncertainty principle.

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