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SHAPIRO’S UNCERTAINTY PRINCIPLE IN THE DUNKL SETTING

Part of: Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  29 April 2015

SAIFALLAH GHOBBER
SAIFALLAH GHOBBER*
Affiliation:
Université de Tunis El Manar, Faculté des Sciences de Tunis, LR11ES11 Analyse Mathématiques et Applications, 2092, Tunis, Tunisie email Saifallah.Ghobber@math.cnrs.fr, Saifallah.Ghobber@ipein.rnu.tn
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Abstract

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The Dunkl transform ${\mathcal{F}}_{k}$ is a generalisation of the usual Fourier transform to an integral transform invariant under a finite reflection group. The goal of this paper is to prove a strong uncertainty principle for orthonormal bases in the Dunkl setting which states that the product of generalised dispersions cannot be bounded for an orthonormal basis. Moreover, we obtain a quantitative version of Shapiro’s uncertainty principle on the time–frequency concentration of orthonormal sequences and show, in particular, that if the elements of an orthonormal sequence and their Dunkl transforms have uniformly bounded dispersions then the sequence is finite.

Keywords

Dunkl transformShapiro’s uncertainty principlephase space localisationtime–frequency concentration

MSC classification

Secondary: 42C20: Other transformations of harmonic type
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 1 , August 2015 , pp. 98 - 110
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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