Localization Operators Associated with the Hypergeometric Wigner Transform Related to the Cherednik Operators in the Cas
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Localization Operators Associated with the Hypergeometric Wigner Transform Related to the Cherednik Operators in the Case of the Root System BCd Amina Hassini1 · Hatem Mejjaoli2 · Khalifa Trimèche1 Received: 1 April 2019 / Revised: 27 November 2019 / Accepted: 4 December 2019 © Iranian Mathematical Society 2020
Abstract Localization operators have a relatively recent development in pure and applied mathematics. Motivated by Wong’s approach, we will study in this paper the time–frequency analysis associated with the hypergeometric Wigner transform related to the Cherednik operators in the case of the root system BCd . Keywords Cherednik operators · Hypergeometric Wigner transform · Localization operators · Root system BCd Mathematics Subject Classification 33E30 · 51F15 · 33C67 · 43A32
1 Introduction We consider the differential–difference operators T j , j = 1, 2, ..., d, associated with a root system R, and a multiplicity function k, introduced by Cherednik in [4], and called the Cherednik operators in the literature. These operators were helpful for the extension
Dedicated to the spirit of Ali Hassini. Communicated by Hamid Reza Ebrahimi Vishki.
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Hatem Mejjaoli [email protected] Amina Hassini [email protected] Khalifa Trimèche [email protected]
1
Department of Mathematics, University of Tunis El Manar Faculty of Sciences of Tunis, Campus, 2092 Tunis, Tunisia
2
Department of Mathematics, College of Sciences, Taibah University, PO BOX 30002, Al Madinah AL Munawarah, Saudi Arabia
123
Bulletin of the Iranian Mathematical Society
and simplification of the theory of Heckman–Opdam which is a generalization of the harmonic analysis on the symmetric spaces G/K , ( [18,19,21]). The Cherednik and Heckman–Opdam theories are based on the Opdam–Cherednik hypergeometric function G λ , λ ∈ Cd , which is the unique analytic solution of the system: T j u(x) = −iλ j u(x),
j = 1, 2, ..., d,
satisfying the normalizing condition u(0) = 1, and the Heckman–Opdam kernel Fλ , λ ∈ Cd , which is defined by: ∀ x ∈ Rd ,
Fλ (x) =
1 G λ (wx), |W | w∈W
where W is the Weyl group associated with the root system R, ( [18,19]). With the kernel G λ , Opdam and Cherednik have defined in [4,18] the Opdam– Cherednik transform Hk and have used the kernel Fλ to define the hypergeometric Fourier transform HkW on spaces of W -invariant functions, and have established some of their properties (see also [19]). Very recently, many authors have been investigating the behavior of the hypergeometric Fourier transform to several problems already studied for the Fourier transform; for instance, hypergeometric Wigner transform [5], real Paley–Wiener theorems [15], uncertainty principles [16], Ramanujan’s Master theorem [17], heat equation [21], and so on. One of the aims of the Fourier transform is the study of the theory of localization operators. This theory was initiated by Daubechies in [6–8], developed in the paper [10] by He and Wong, and detailed in the book [26] by Wong. We emphasize the fact that localization operators
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