Uncertainty Principles for the Dunkl-Wigner Transforms

Fethi Soltani

doi:10.1155/2016/7637346

Abstract

We prove a version of Heisenberg-type uncertainty principle for the Dunkl-Wigner transform of magnitude s>0; and we deduce a local uncertainty principle for this transform.

Highlights

In this paper, we consider Rd with the Euclidean inner product ⟨⋅, ⋅⟩ and norm |y| fl √⟨y, y⟩
Many uncertainty principles have already been proved for the Dunkl transform Fk, namely, by Rosler [2] and Shimeno [3] who established the Heisenberg-type uncertainty inequality for this transform, by showing that for f ∈ L2(μk)
(−t)dμk (t) where gk,y (z) fl Fk (√τy 󵄨󵄨󵄨󵄨Fk (g)󵄨󵄨󵄨󵄨2) (z). This transform is studied in [9, 10] where the author established some applications (Plancherel formula, inversion formula, Calderon’s reproducing formula, extremal function, etc.)

Summary

Introduction

We consider Rd with the Euclidean inner product ⟨⋅, ⋅⟩ and norm |y| fl √⟨y, y⟩. Many uncertainty principles have already been proved for the Dunkl transform Fk, namely, by Rosler [2] and Shimeno [3] who established the Heisenberg-type uncertainty inequality for this transform, by showing that for f ∈ L2(μk),. The author [4,5,6,7] proved general forms of the Heisenberg-type inequality for the Dunkl transform Fk. Let g ∈ L2rad(μk). Where gk,y (z) fl Fk (√τy 󵄨󵄨󵄨󵄨Fk (g)󵄨󵄨󵄨󵄨2) (z) This transform is studied in [9, 10] where the author established some applications (Plancherel formula, inversion formula, Calderon’s reproducing formula, extremal function, etc.). We prove a Heisenberg-type uncertainty principle for the Dunkl-Wigner transform Vg of magnitude s > 0; that is, there exists a constant c(k, s) > 0 such that, for f ∈ L2(μk),.

The Dunkl-Wigner Transform

Uncertainty Principles for the Mapping Vg