Table of Contents
Journal of Operators
Volume 2016, Article ID 7637346, 7 pages
http://dx.doi.org/10.1155/2016/7637346
Research Article

Uncertainty Principles for the Dunkl-Wigner Transforms

Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 277, Jazan 45142, Saudi Arabia

Received 31 July 2016; Accepted 6 September 2016

Academic Editor: Dashan Fan

Copyright © 2016 Fethi Soltani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [47] 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.

Theorem 8. Let be a nonzero function and . Then, for , one has

Proof. Let and let , , such that . Then, for , we have where is defined as usual by . By Hölder’s inequality we get Thus, for all , we have with equality if . In the same manner and using Theorem 1(iii), we have, for , with equality if . By (48) and (49), for all , we have with equality if . Applying Theorem 7, we obtain which completes the proof of the theorem.

From (48) and (49) we deduce the following remark.

Remark 9. Let be a nonzero function and . Then, for , we have

For , we define the dilation of by Then

Let us now turn to establishing Heisenberg-type uncertainty principle for the Dunkl-Wigner transform of magnitude . Thus, we consider the following lemma.

Lemma 10. Let and let be a nonzero function. Then, for , one has

Proof. From Proposition 4(ii), we have But by (55) we have Thus, which gives the result.

Theorem 11 (Heisenberg-type uncertainty principle for ). Let . Then there exists a constant such that, for all and , one has

Proof. Let and , where . Fix such that . We write But from Hölder’s inequality and Proposition 4(iii) we have Therefore, by Theorem 5(i),Using the fact that we deduce thatwhere Replacing and by and , respectively, in the previous inequality, we obtain by Lemma 10 and by a suitable change of variables By setting in the right-hand side of the previous inequality we obtain the desired result.

We will now prove a local uncertainty principle for the Dunkl-Wigner transform , which extends the result of Faris [22].

Theorem 12 (local uncertainty principle for ). Let . Then there exists a constant such that, for all and and for all measurable subset of such that , one has

Proof. Let and let be a measurable subset of such that . From Hölder’s inequality and Proposition 4(iii) we have From (63) there exists such that Therefore we obtain the desired result.

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

  1. C. F. Dunkl, “Hankel transforms associated to finite reection groups,” Contemporary Mathematics, vol. 138, pp. 123–138, 1992. View at Publisher · View at Google Scholar
  2. M. Rösler, “An uncertainty principle for the Dunkl transform,” Bulletin of the Australian Mathematical Society, vol. 59, no. 3, pp. 353–360, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. N. Shimeno, “A note on the uncertainty principle for the Dunkl transform,” The University of Tokyo. Journal of Mathematical Sciences, vol. 8, no. 1, pp. 33–42, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. F. Soltani, “Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform on Rd,” Bulletin of the Australian Mathematical Society, vol. 87, no. 2, pp. 316–325, 2013. View at Publisher · View at Google Scholar
  5. F. Soltani, “A general form of Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform,” Integral Transforms and Special Functions, vol. 24, no. 5, pp. 401–409, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. F. Soltani, “An Lp Heisenberg-Pauli-Weyl uncertainty principle for the Dunkl transform,” Konuralp Journal of Mathematics, vol. 2, no. 1, pp. 1–6, 2014. View at Google Scholar
  7. F. Soltani, “Uncertainty principles and extremal functions for the Dunkl L2-multiplier operators,” Journal of Operators, vol. 2014, Article ID 659069, 9 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  8. S. Thangavelu and Y. Xu, “Convolution operator and maximal function for the Dunkl transform,” Journal d'Analyse Mathématique, vol. 97, pp. 25–55, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. F. Soltani, “Inversion formula for the Dunkl-Wigner transform and compactness property for the Dunkl-Weyl transforms,” Journal of Mathematical Research with Applications, vol. 35, no. 4, pp. 425–434, 2015. View at Google Scholar
  10. F. Soltani, “Reproducing inversion formulas for the Dunkl-Wigner transforms,” Cubo: A Mathematical Journal, vol. 17, no. 2, pp. 1–14, 2015. View at Google Scholar
  11. H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, New York, NY, USA, 1950.
  12. M. W. Wong, Weyl Transforms, Universitext, Springer, New York, NY, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  13. A. Dachraoui, “Weyl-Bessel transforms,” Journal of Computational and Applied Mathematics, vol. 133, no. 1-2, pp. 263–276, 2001. View at Google Scholar
  14. C. F. Dunkl, “Integral kernels with reflection group invariance,” Canadian Journal of Mathematics, vol. 43, no. 6, pp. 1213–1227, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  15. M. F. E. de Jeu, “The Dunkl transform,” Inventiones Mathematicae, vol. 113, no. 1, pp. 147–162, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. E. M. Opdam, “Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group,” Compositio Mathematica, vol. 85, no. 3, pp. 333–373, 1993. View at Google Scholar · View at MathSciNet
  17. M. Rösler, “Positivity of Dunkl's intertwining operator,” Duke Mathematical Journal, vol. 98, no. 3, pp. 445–463, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. M. Rösler and M. Voit, “Markov processes related with Dunkl operators,” Advances in Applied Mathematics, vol. 21, no. 4, pp. 575–643, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. G. N. Watson, A Treatise on Theory of Bessel Functions, Cambridge University Press, Cambridge, UK, 1966.
  20. M. Rösler, “A positive radial product formula for the Dunkl kernel,” Transactions of the American Mathematical Society, vol. 355, no. 6, pp. 2413–2438, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. S. Thangavelu and Y. Xu, “Riesz transform and Riesz potentials for Dunkl transform,” Journal of Computational and Applied Mathematics, vol. 199, no. 1, pp. 181–195, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. W. G. Faris, “Inequalities and uncertainty inequalities,” Journal of Mathematical Physics, vol. 19, pp. 461–466, 1978. View at Google Scholar