Abstract

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

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 Coxeter group is a subgroup of the orthogonal group . 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 (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 , by , , the space of measurable functions on , such that and by the subspace of consisting of radial functions.

For the Dunkl transform of is defined (see [1]) by where denotes the Dunkl kernel. (For more details see the next section.)

Many uncertainty principles have already been proved for the Dunkl transform , namely, by Rösler [2] and Shimeno [3] who established the Heisenberg-type uncertainty inequality for this transform, by showing that for , Recently, the author [4–7] proved general forms of the Heisenberg-type inequality for the Dunkl transform .

The Dunkl translation operators , , [8] are defined on by

Let . The Dunkl-Wigner transform is the mapping defined for by where This transform is studied in [9, 10] where the author established some applications (Plancherel formula, inversion formula, Calderón’s reproducing formula, extremal function, etc.).

In this paper we use formula (4); we prove uncertainty principle intervening and of magnitudes ; that is, for every ,Next, we prove a Heisenberg-type uncertainty principle for the Dunkl-Wigner transform of magnitude ; that is, there exists a constant such that, for , Finally, we prove a local uncertainty principle for the Dunkl-Wigner transform ; that is, there exists a constant such that, for and for measurable subset of such that , where is the indicator function of the set .

In the classical case, the Fourier-Wigner transforms are studied by Weyl [11] and Wong [12]. In the Bessel-Kingman hypergroups, these operators are studied by Dachraoui [13].

This paper is organized as follows. In Section 2, we recall some properties of the Dunkl-Wigner transform . In Section 3, we prove a Heisenberg-type uncertainty principle for the Dunkl-Wigner transform of magnitude ; and we deduce a local uncertainty principle for this transform.

2. The Dunkl-Wigner Transform

The Dunkl operators , , on associated with the finite reflection group and multiplicity function are given, for a function of class on , by

For , the initial value problem , , with admits a unique analytic solution on , which will be denoted by and called Dunkl kernel [14, 15]. This kernel has a unique analytic extension to (see [16]). The Dunkl kernel has the Laplace-type representation [17] where and is a probability measure on , such that . In our case,

The Dunkl kernel gives rise to an integral transform, which is called Dunkl transform on , and was introduced by Dunkl in [1], where already many basic properties were established. Dunkl’s results were completed and extended later by de Jeu [15]. The Dunkl transform of a function in is defined by We notice that agrees with the Fourier transform that is given by

The Dunkl transform of a function which is radial is again radial and could be computed via the associated Fourier-Bessel transform (see [18], Proposition 4); that is, where and Here is the spherical Bessel function (see [19]).

Some of the properties of Dunkl transform are collected below (see [1, 15]).

Theorem 1. (i) -Boundedness. For all , , and (ii) Inversion Theorem. Let , such that . Then (iii) Plancherel Theorem. The Dunkl transform extends uniquely to an isometric isomorphism of onto itself. In particular, one has (iv) Parseval Theorem. For , one has

The Dunkl transform allows us to define a generalized translation operators on by settingIt is the definition of Thangavelu and Xu given in [8]. It plays the role of the ordinary translation in , since the Euclidean Fourier transform satisfies . Note that, from (13) and Theorem 1(iii), relation (22) makes sense, and , for all .

Rösler [20] introduced the Dunkl translation operators for radial functions. If are radial functions, , then where is the representing measure given by (12).

This formula allows us to establish the following results [8, 21].

Proposition 2. (i) For all and for all , the Dunkl translation is a bounded operator, and for , one has (ii) Let . Then, for all , one has

The Dunkl convolution product of two functions and in is defined by We notice that generalizes the convolution that is given by

Proposition 2 allows us to establish the following properties for the Dunkl convolution on (see [8]).

Proposition 3. (i) Assume that and such that . Then the map extends to a continuous map from to , and (ii) For all and , one has (iii) Let and . Then belongs to if and only if belongs to , and (iv) Let and . Then where both sides are finite or infinite.

Let and . The modulation of by is the function defined by Thus,

Let . The Fourier-Wigner transform associated with the Dunkl operators is the mapping defined for by

In the following we recall some properties of the Dunkl-Wigner transform (Plancherel formula, inversion formula, reproducing inversion formula of Calderón’s type, etc.).

Proposition 4 (see [10]). Let . Then(i).(ii).(iii)The function belongs to , and

Theorem 5 (see [10]). Let be a nonzero function. Then one has the following.(i) Plancherel formula: for every , one has (ii) Parseval formula: for every , one has (iii) Inversion formula: for all such that , one has

Theorem 6 (Calderón’s reproducing inversion formula; see [10]). Let , , and let be a nonzero function, such that . Then, for , the function given by belongs to and satisfies

3. Uncertainty Principles for the Mapping

In this section we establish Heisenberg-type uncertainty principle for the Dunkl-Wigner transform . We begin by the following theorem.

Theorem 7. Let be a nonzero function. Then, for , one has

Proof. Let . Assume that . Inequality (4) leads to Integrating with respect to and using the Schwarz inequality, we get But by Proposition 4(ii), Fubini-Tonelli’s theorem, (16), Proposition 2(ii), and Theorem 1(iii), we have This yields the result and completes the proof of the theorem.