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Clifford Fourier transformation and uncertainty principle for the Clifford geometric algebra \(\mathrm{Cl}_{3,0}\). (English) Zbl 1133.42305

Summary: First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions \((f:\mathbb{R}^3 \to \mathrm{Cl}_{3,0})\). Third, we show a set of important properties of the Clifford Fourier transform on \(\mathrm{Cl}_{3,0}\) such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for \(\mathrm{Cl}_{3,0}\) multivector functions.

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
15A66 Clifford algebras, spinors
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