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).
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
( 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
Fourth, we develop and utilize commutation properties
for giving explicit formulas (*) for f x^m , f Delta^m and for the Clifford convolution.
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.
[ PDF ] 316K (Updated 9 May 2008)