E.M.S. Hitzer, B. Mawardi

*Cl*(3,0)
based on Clifford Fourier Transform

in TE. Simos, G. Sihoyios, C. Tsitouras (eds.),
*International Conference on Numerical Analysis and Applied
Mathematics 2005*, Wiley-VCH, Weinheim, 2005, pp. 922-925.

**Introduction:**
In the field of applied mathematics the Fourier transform has developed into an important tool. It is a powerful
method for solving partial differential equations. The Fourier transform provides also a technique for signal
analysis where the signal from the original domain is transformed to the spectral or frequency domain. In the
frequency domain many characteristics of the signal are revealed. With these facts in mind, we extend the Fourier
transform in geometric algebra.

Brackx et al. [1] extended the Fourier transform to multivector valued function-distributions in *Cl*(0,n) with
compact support. They also showed some properties of this generalized Fourier transform. A related applied
approach for hypercomplex Clifford Fourier Transformations in *Cl*(0,n)
was followed by Buelow et. al. [2]. In [3],
Li et. al. extended the Fourier Transform holomorphically to a function of m complex variables.

In this paper we adopt and expand the generalization of the Fourier transform in Clifford geometric algebra
*G*_3 recently suggested by Ebling and Scheuermann [5]. We explicitly show detailed properties of the real Clifford
geometric algebra Fourier transform (CFT), which we subsequently use to define and prove the uncertainty
principle for *G*_3 multivector functions.

**Key words:**
vector derivative, multivector-valued function, Clifford (geometric) algebra, Clifford Fourier transform, uncertainty principle.

**Subject classification:** 15A66

[PDF] 107kb

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