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Two-Wavelet Multipliers on the Dual of the Laguerre Hypergroup and Applications

  • Hatem MejjaoliEmail author
  • Khalifa Trimèche
Article
  • 27 Downloads

Abstract

In this paper, we are interested in the Laguerre hypergroup \(\mathbb {K} = [0,\infty )\times {\mathbb {R}}\) which is the fundamental manifold of the radial function space for the Heisenberg group. So, we consider the generalized shift operator generated by the dual of the Laguerre hypergroup \(\widehat{\mathbb {K}}\) which can be topologically identified with the so-called Heisenberg fan, the subset of \({\mathbb {R}}^{2}\)
$$\begin{aligned} \cup _{j\in {\mathbb {N}}}\left\{ (\lambda ,\mu )\in {\mathbb {R}}^{2}:\mu =|\lambda |(2j+\alpha +1), \; \lambda \ne 0\right\} \cup \left\{ (0,\mu )\in {\mathbb {R}}^{2}:\mu \ge 0\right\} , \end{aligned}$$
by means of which the notion of a generalized two-wavelet multiplier is investigated. The boundedness and compactness of the generalized two-wavelet multipliers are studied on \(L^{p}_{\alpha }(\mathbb {K})\), \(1 \le p \le \infty \). Afterwards, we introduce the generalized Landau–Pollak–Slepian operator and we give its trace formula. We show that the generalized two-wavelet multiplier is unitary equivalent to a scalar multiple of the generalized Landau–Pollak–Slepian operator. As applications, we prove an uncertainty principle of Donoho–Stark type involving \(\varepsilon \)-concentration of the generalized two-wavelet multiplier operators. Moreover, we study functions whose time–frequency content is concentrated in a region with finite measure in phase space using the phase space restriction operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators.

Keywords

Laguerre hypergroup generalized multipliers generalized two-wavelet multipliers Schatten–von Neumann class generalized Landau–Pollak–Slepian operator 

Mathematics Subject Classification

33E30 43A32 81S30 94A12 45P05 42C25 42C40 

Notes

Acknowledgements

The authors are deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article. The first author thanks the professor M.W. Wong for his help.

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Authors and Affiliations

  1. 1.Department of MathematicsCollege of Sciences, Taibah UniversityAl Madinah Al MunawarahSaudi Arabia
  2. 2.Department of MathematicsFaculty of Sciences of Tunis, University of Tunis El ManarTunisTunisia

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