Time–Frequency Localization Operators and a Berezin Transform

Journal de Mathématiques Pures et Appliquées

2018

Quantum harmonic analysis on phase space is shown to be linked with localization operators. The convolution between operators and the convolution between a function and an operator provide a conceptual framework for the theory of localization operators which is complemented by an appropriate Fourier transform, the Fourier-Wigner transform. We link the Hausdorff-Young inequality for the Fourier-Wigner transform with Lieb's inequality for ambiguity functions. Noncommutative Tauberian theorems due to Werner allow us to extend results of Bayer and Gröchenig on localization operators. Furthermore we show that the Arveson spectrum and the theory of Banach modules provide the abstract setting of quantum harmonic analysis.1991 Mathematics Subject Classification. 47G30; 35S05; 46E35; 47B10.

Section: Remarkmentioning

confidence: 99%

“…For any regular T 0 ∈ T 1 , BT is p-regular. The density in points (1), (3) and (5) is in the p norm for p < ∞, and weak * density for p = ∞.…”

Section: 1mentioning

confidence: 99%

Convolutions for localization operators

Journal de Mathématiques Pures et Appliquées

2018

“…If we pick S = ϕ 2 ⊗ϕ 1 for ϕ 1 , ϕ 2 ∈ L 2 (R d ) in the three previous propositions, the conditions on the set of zeros of F W (S) becomes a condition on the zeros of the ambiguity function A(ϕ 2 , ϕ 1 ). We noted this in [55], where we generalized previous results from [4]. For such rank-one operators, proposition 5.3 raises a natural question: Does there exist a pair of windows ϕ 1 , ϕ 2 ∈ L 2 (R d ) such that A(ϕ 2 , ϕ 1 ) has no zeros, except when ϕ 1 = ϕ 2 is a Gaussian?…”

Section: 2mentioning

confidence: 67%

Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators

2019

J Fourier Anal Appl

We study mixed-state localization operators from the perspective of Werner's operator convolutions which allows us to extend known results from the rank-one case to trace class operators. The idea of localizing a signal to a domain in phase space is approached from various directions such as bounds on the spreading function, probability densities associated to mixed-state localization operators, positive operator valued measures, positive correspondence rules and variants of Tauberian theorems for operator translates. Our results include a rigorous treatment of multiwindow-STFT filters and a characterization of mixed-state localization operators as positive correspondence rules. Furthermore we provide a description of the Cohen class in terms of Werner's convolution of operators and deduce consequences on positive Cohen class distributions, an uncertainty principle, uniqueness and phase retrieval for general elements of Cohen's class.where π(z)ψ(t) = e 2πiωt ψ(t−x). In [55] we showed that this yields a natural class of Banach modules. There are two types of convolutions in this noncommutative setting: (i) The convolution between a function f ∈ L 1 (R 2d ) and a trace class operator S:(ii) the convolution between two trace class operators S and T is defined by1991 Mathematics Subject Classification. 47G30; 35S05; 46E35; 47B10. Key words and phrases. localization operators, Cohen class, uncertainty principle, phase retrieval, positive operator valued measures.F W S(z) = e −πix·ω tr(π(−z)S) for z ∈ R 2d . Note that the Fourier-Wigner transform and the spreading function differ only by a phase factor. The Fourier-Wigner transform has many properties analogous to those of the Fourier transform of functions [55,66]. In the case of rank-one operators these concepts of quantum harmonic analysis turn into wellknown objects from time-frequency analysis. Suppose ϕ 2 ⊗ ϕ 1 is the rank-one operator for ϕ 1 , ϕ 2 ∈ L 2 (R d ). Then we havewhich is a localization operator (or STFT-filter or 46]) and is denoted by A ϕ1,ϕ2f , and f is called the mask of the STFT-filter. Similarly, the convolution of two rank-one operators becomeswhereξ(x) = ξ(−x), which reduces for η = ψ and ψ = φ to the spectrogram [42]. The Fourier-Wigner transform of a rank-one operator is the ambiguity function. There is also a Hausdorff-Young inequality associated to the Fourier-Wigner transform [55,66], that in the rank-one case is the non-sharp Lieb's inequality for ambiguity functions [51]. Let us return to the objectives of this paper. Since localization operators are convolutions of a function and a rank-one operator, a natural extension of localization operators are operators of the form f ⋆ S for a trace-class operator S. The results of this paper indicate that these operators describe the time-frequency localization in various ways. For example we are interested in the amount of "spreading" in time and frequency that an operator performs on a function which we describe in form of bounds on the concentration of the spreading function, or equivalently ...

“…Weyl operator e 2πiωQ−ixP [14,15] Integrated Schrödinger representation Weyl quantization [15] Twisted Weyl symbol, Fourier-Wigner transform Weyl representation [15] Glauber-Sudarshan representation anti-Wick symbol [5], contravariant Berezin symbol, upper symbol [20], symbol for localization operator [1] Husimi representation Berezin transform [1], covariant Berezin symbol, lower symbol [20] Table 1. A dictionary relating the terminology in this paper to other common terminologies in mathematical physics.…”

Section: This Papermentioning

confidence: 99%

Convolutions for Berezin quantization and Berezin-Lieb inequalities

Journal of Mathematical Physics

2018

Concepts and results from quantum harmonic analysis, such as the convolution between functions and operators or between two operators, are identified as the appropriate setting for Berezin quantization and Berezin-Lieb inequalities. Based on this insight we provide a rigorous approach to generalized phase-space representation introduced by Klauder-Skagerstam and their variants of Berezin-Lieb inequalities in this setting. Hence our presentation of the results of Klauder-Skagerstam gives a more conceptual framework, which yields as a byproduct an interesting perspective on the connection between Berezin quantization and Weyl quantization. 1991 Mathematics Subject Classification. 47G30; 35S05; 46E35; 47B10.