Shapiro’s Uncertainty Principle in the Dunkl setting

Abstract

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.

Logarithm Sobolev and Shannon’s Inequalities Associated with the Deformed Fourier Transform and Applications

“…In this section, we revisit and prove new Heisenberg-type uncertainty principles for the Dunkl transform. The first result is the following well-known uncertainty inequality (see [8][9][10]).…”

“…However, the proof of Heisenberg-type uncertainty inequality (10) can be obtained from either the Faris-type local uncertainty principle [7] (Theorem A) or from the Benedicks-Amrein-Berthier uncertainty principle [7] (Theorem B). Moreover, the sharp constant in (10) is well known only for the special case s = β = 1, and it is equal to γ + d/2; equality in (10) occurs if and only if f (x) = ce −λ|x| 2 for some c ∈ C, and λ > 0 (see [8][9][10]).…”

Fourier-Like Multipliers and Applications for Integral Operators

“…[13]), the the Dunkl transform (see e.g. [14]), the G-transform (see e.g. [29]), the deformed Fourier transform (see e.g.…”