$$L^{p}$$ L p local uncertainty principles for the Dunkl Gabor transform on $${\mathbb {R}}^{d}$$ R d
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ORIGINAL ARTICLE
Lp local uncertainty principles for the Dunkl Gabor transform on Rd Hatem Mejjaoli1 • Nadia Sraieb2 • Khalifa Trime`che3 Received: 20 September 2019 / Accepted: 5 June 2020 Ó Sociedad Matemática Mexicana 2020
Abstract The purpose of this paper is to establish the Lp local uncertainty principles for the Dunkl Gabor transform on Rd . These allow us to prove a couple of global uncertainty inequalities. Keywords Dunkl Gabor transform Local uncertainty principles
Mathematics Subject Classification 42B10 44A05
1 Introduction We consider the differential-difference operators Tj , j ¼ 1; 2; . . .; d, associated with a root system R and a multiplicity function k, introduced by Dunkl in [4], and called the Dunkl operators in the literature. The Dunkl theory is based on the Dunkl kernel Kðk; :Þ; k 2 Cd , which is the unique analytic solution of the system & Hatem Mejjaoli [email protected] Nadia Sraieb [email protected] Khalifa Trime`che [email protected] 1
Department of Mathematics, College of Sciences, Taibah University, P.O. Box 30002, Al Madinah AL Munawarah, Saudi Arabia
2
Department of Mathematics, Faculty Sciences of Gabe`s, University of Gabe`s, Riadh Zerig 6029, Gabe`s, Tunisia
3
Department of Mathematics, Faculty of Sciences of Tunis, University of El Manar, Campus, 2092 Tunis, Tunisia
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Tj uðxÞ ¼ kj uðxÞ;
j ¼ 1; 2; . . .; d;
satisfying the normalizing condition uð0Þ ¼ 1. With the kernel Kðk; :Þ, Dunkl have defined in [5] the Dunkl transform F D . For a family of weighted functions, xk , invariant under a finite reflection group W, Dunkl transform is an extension of the Fourier transform that defines an isometry of L2 ðRd ; xk ðxÞdxÞ onto itself. The basic properties of the Dunkl transforms have been studied by several authors, see [3–5, 26] and the references therein. Very recently, many authors have been investigating the behavior of the Dunkl transform to several problems already studied for the Fourier transform; for instance, uncertainty principles [11], real Paley–Wiener theorems [13], heat equation [20], Dunkl wavelet transform [27], and so on. One of the aims of the Fourier transform is the study of the Gabor transform. This transform originates from the work of Dennis Gabor [9], in which he used translations and modulations of a single Gaussian to represent one dimensional signals. Yet there are still several gaps in our knowledge of Dunkl harmonic analysis. One of the main reasons is the lack of tools related to the generalized translation operator. Unfortunately, the Lp -boundedness and the positivity of this generalized translation operator are not obtained in general. At the moment, an explicit formula for the generalized translation operator is known only in two cases: when the function is radial and when the finite reflection group W ¼ Zd2 (See [22, 25]). As the positivity of the generalized translation operator is not obtained in general, we have defined and studied the Dunkl Gabor transform on Rd , in which we used the generalize
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