Convolutions for Berezin quantization and Berezin-Lieb inequalities

Luef, Franz; Skrettingland, Eirik

doi:10.1063/1.5023241

Cited by 8 publications

(4 citation statements)

References 23 publications

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“…The goal of this paper is therefore to develop a version of quantum harmonic analysis for lattices to provide a similar conceptual framework for Gabor multipliers. Hence we continue the line of research into applications of quantum harmonic analysis from [45][46][47].…”

Section: Introductionmentioning

confidence: 99%

Quantum Harmonic Analysis on Lattices and Gabor Multipliers

Skrettingland

2020

J Fourier Anal Appl

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We develop a theory of quantum harmonic analysis on lattices in R 2d . Convolutions of a sequence with an operator and of two operators are defined over a lattice, and using corresponding Fourier transforms of sequences and operators we develop a version of harmonic analysis for these objects. We prove analogues of results from classical harmonic analysis and the quantum harmonic analysis of Werner, including Tauberian theorems and a Wiener division lemma. Gabor multipliers from time-frequency analysis are described as convolutions in this setting. The quantum harmonic analysis is thus a conceptual framework for the study of Gabor multipliers, and several of the results include results on Gabor multipliers as special cases.Keywords Gabor multipliers • Tauberian theorems • Feichtinger's algebra • Fourier-Wigner transformCommunicated by Hans G. Feichtinger.

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Section: Introductionmentioning

confidence: 99%

Quantum Harmonic Analysis on Lattices and Gabor Multipliers

Skrettingland

2020

J Fourier Anal Appl

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“…A Berezin-Lieb inequality. The Berezin-Lieb inequality as investigated in [40,46,58] can be seen as a generalization of Corollary 4.4. We present here a generalization to locally compact groups with the proof partially based on a proof for the Weyl-Heisenberg case in [22].…”

Section: 5mentioning

confidence: 98%

Quantum harmonic analysis on locally compact groups

Simon¹

2022

Preprint

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On a locally compact group we introduce covariant quantization schemes and analogs of phase space representations as well as mixed-state localization operators. These generalize corresponding notions for the affine group and the Heisenberg group. The approach is based on associating to a square integrable representation of the locally compact group two types of convolutions between integrable functions and trace class operators. In the case of non-unimodular groups these convolutions only are well-defined for admissible operators, which is an extension of the notion of admissible wavelets as has been pointed out recently in the case of the affine group.

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“…1.70]. Zero-free Wigner distributions occur prominently in [26,28] in a similar context. It is therefore natural to ask for examples that satisfy Bayer's assumptions:…”

Section: Introductionmentioning

confidence: 96%

“…1.70]. Zero-free Wigner distributions occur prominently in [26,28] in a similar context. It is therefore natural to ask for examples that satisfy Bayer's assumptions: e −q ∈ L 2 (R d ) (generalized Gaussian), then the Wigner distribution W (f, g) is also zerofree.…”

Section: Introductionmentioning

confidence: 96%

Zeros of the Wigner distribution and the short-time Fourier transform

2019

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We study the question under which conditions the zero set of a (cross-) Wigner distribution W (f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson's theorem for the positivity of the Wigner distribution and to Hardy's uncertainty principle. We then construct a class of step functions S so that the Wigner distribution W (f, 1 (0,1) ) always possesses a zero f ∈ S ∩ L p for p < ∞, but may be zero-free for f ∈ S ∩ L ∞ . The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.

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Convolutions for Berezin quantization and Berezin-Lieb inequalities

Cited by 8 publications

References 23 publications

Quantum Harmonic Analysis on Lattices and Gabor Multipliers

Quantum Harmonic Analysis on Lattices and Gabor Multipliers

Quantum harmonic analysis on locally compact groups

Zeros of the Wigner distribution and the short-time Fourier transform

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