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.

[ PDF ] 316K (Updated 9 May 2008)

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