Gabor multipliers associated with the Bessel–Kingman hypergroup

Abstract

We consider the Bessel–Kingman hypergroup ([0,+∞[,∗) and we denote by dμα,α⩾-12,the measure defined on [0,+∞[ by dμα(x)=x2+12αΓ(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 dμα called window functions and σ is 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,+∞].

Full text

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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