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J. Pseudo-Differ. Oper. Appl. (2021) 12:45

https://doi.org/10.1007/s11868-021-00415-w

Gabor multipliers associated with the Bessel–Kingman

hypergroup

Lakhdar T. Rachdi1·Besma Amri1

Received: 4 April 2021 / Revised: 12 July 2021 / Accepted: 16 July 2021 /

Published online: 27 July 2021

© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021

Abstract

We consider the Bessel–Kingman hypergroup [0,+∞[,∗and we denote by

dμα,α≥−

1

2,the measure deﬁned on [0,+∞[ by dμα(x)=x2+1

2α(α+1)dx.We

deﬁne the Gabor multiplier Gu,v (σ ) associated with two square integrable functions

u,v on [0,+∞[ with respect to the measure dμαcalled window functions and σis a

measurable function on [0,+∞[ called a signal. We prove that Gu,v(σ ) is a bounded

linear operator on L2(dμα)and it is compact. Next, we deﬁne the Schatten von-

Neumann class Sp,p∈[1,+∞],and we show that the Gabor multiplier Gu,v(σ )

belongs to the class Sp.We give also a formula of trace when σ∈L1(dμα).Also,we

deﬁne the Landau–Pollak–Slebian operator and we give its connection with the Gabor

multiplier. Last, we study the boundedness and compactness of the Gabor multipliers

for more window functions u,v ∈Lp(dμα), p∈[1,+∞].

Keywords Bessel–Kingman hypergroup ·Gabor multiplier ·Landau–Pollak–Slebian

operator ·Schatten von-Neumann class ·Hilbert Schmidt operator ·Compact

operator ·Class of trace

Mathematics Subject Classiﬁcation 42A38 ·44A35

1 Introduction

Gabor multipliers, called also localization operators, Toeplitz operators or anti-Wick

operators, were introduced ﬁrstly by Daubechies [11,12] in time frequency analysis.

BLakhdar T. Rachdi

lakhdartannech.rachdi@fst.rnu.tn; lakhdar.rachdi@fst.utm.tn

Besma Amri

besmaa.amri@gmail.com

1Faculté des Sciences de Tunis, LR18ES09 Modélisation Mathématique, Analyse Harmonique et

Théorie du Potentiel, Université de Tunis El Manar, 2092 Tunis, Tunisia

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