E.M.S. Hitzer, B. Mawardi
Uncertainty Principle for the Clifford Geometric Algebra 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


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