Uncertainty Principles and Extremal Functions for the Dunkl -Multiplier Operators

We study some class of Dunkl -multiplier operators; and related to these operators we establish the Heisenberg-Pauli-Weyl uncertainty principle and Donoho-Stark’s uncertainty principle. We give also an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev-Dunkl spaces. 1. Introduction In this paper, we consider with the Euclidean inner product and norm . For , let be the reflection in the hyperplane orthogonal to : A finite set is called a root system, if . ？ and for all . We assume that it is normalized by for all . For a root system , the reflections ,？？ , generate a finite group , the reflection group associated with . All reflections in correspond to suitable pairs of roots. For a given , we fix the positive subsystem . Then for each either or . Let be a multiplicity function on (i.e., a function which is constant on the orbits under the action of ). As an abbreviation, we introduce the index . Throughout this paper, we will assume that for all . Moreover, let denote the weight function , for all , which is -invariant and homogeneous of degree . Let be the Mehta-type constant given by We denote by the measure on given by and by , , the space of measurable functions on , such that For the Dunkl transform is defined (see [1]) by where denotes the Dunkl kernel (for more details, see Section 2). Many uncertainty principles have already been proved for the Dunkl transform, namely, by R？sler [2] and Shimeno [3] who established the Heisenberg-Pauli-Weyl inequality for the Dunkl transform, by showing that, for every , Recently, the author [4, 5] proved general forms of the Heisenberg-Pauli-Weyl inequality for the Dunkl transform. Let be a function in . The Dunkl -multiplier operators, , are defined, for regular functions on , by These operators are studied in [6, 7] where the author established some applications (Calderón’s reproducing formulas, best approximation formulas, and extremal functions…). For satisfying the admissibility condition: , a.e. ？ , then the operators satisfy Plancherel’s formula: where is the measure on given by . For the operators we establish a Heisenberg-Pauli-Weyl uncertainty principle. More precisely, we will show, for , provided satisfying , a.e. . Building on the techniques of Donoho and Stark [8], we show a continuous-time principle for the theory. Let be measurable subset of , let be measurable subset of , and let . If is -concentrated on and is -concentrated on (see Section 3 for more details), then provided satisfying , a.e. . Building on the ideas of Saitoh [9, 10], Matsuura et al.