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Some shaper uncertainty principles for multivector-valued functions. (English) Zbl 1359.15019

It is well known that Heisenberg’s uncertainty principle of quantum mechanics has been generalized to higher dimensions to state that a function and its Fourier transform cannot be well localized simultaneously. Indeed, many versions of the uncertainty principle for the Fourier and other transforms have appeared in the literature. In signal processing, multiplexing involves encoding multivector-valued functions without loss of information by means of the Clifford algebra. The Fourier transform has also been extended to apply to multivector-valued functions. In this paper, the authors study the Clifford-Fourier transform and, using operator theory methods, derive a stronger Heisenberg-Pauli-Weyl-type uncertainty principle in this context. In the last section, they present a stronger uncertainty principle for the Dunkl transform, generalizing a result of M. Rösler [Bull. Aust. Math. Soc. 59, No. 3, 353–360 (1999; Zbl 0939.33012)].

MSC:

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

Citations:

Zbl 0939.33012
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References:

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