Convolutions for localization operators

Luef, Franz; Skrettingland, Eirik

doi:10.1016/j.matpur.2017.12.004

Cited by 44 publications

(86 citation statements)

References 29 publications

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Section: 3mentioning

confidence: 99%

“…The convolutions can be defined on other L p -spaces and Schatten p-classes by duality [27,36]. As a special case we mention that (4) defines a continuous function even when T ∈ B(L 2 (R d )) [27]; in particular it is clear from (4) that (5) S ⋆ I(z) = tr(S) for any z ∈ R 2d when I is the identity operator and S ∈ T . The convolutions of operators and functions are associative, a fact that is non-trivial since the convolutions between operators and functions can produce both operators and functions as output [27,36].…”

Section: 3mentioning

confidence: 99%

“…The main object of this paper is an in-depth treatment of the mixed-state localization operators from [29] and their associated time-frequency distributions from the perspective developed in [2,3], and to describe them we first recall some facts about quantum harmonic analysis [27,36]. Concretely, the convolution between two trace class operators and the convolution between a function and a trace class operator.…”

Section: Introductionmentioning

confidence: 99%

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On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators

Luef

Skrettingland

2019

Constr Approx

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Recently we introduced mixed-state localization operators associated to a density operator and a (compact) domain in phase space. We continue the investigations of their eigenvalues and eigenvectors. Our main focus is the definition of a time-frequency distribution which is based on the Cohen class distribution associated to the density operator and the eigenvectors of the mixed-state localization operator. This time-frequency distribution is called the accumulated Cohen class distribution. If the trace class operator is a rank-one operator, then the mixed-state localization operators and the accumulated Cohen class distribution reduce to Daubechies' localization operators and the accumulated spectrogram. We extend all the results about the accumulated spectrogram to the accumulated Cohen class distribution. The techniques used in the case of spectrograms cannot be adapted to other distributions in Cohen's class since they rely on the reproducing kernel property of the shorttime Fourier transform. Our approach is based on quantum harmonic analysis on phase space which also provides the tools and notions to introduce the analogues of the accumulated spectrogram for mixed-state localization operators; the accumulated Cohen's class distributions.1991 Mathematics Subject Classification. 81S30, 94A12, 42C25 .

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Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators

Luef

Skrettingland

2019

Constr Approx

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“…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.…”

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confidence: 99%

“…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.…”

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

Luef

Skrettingland

2019

J Fourier Anal Appl

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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 ...

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Generalized Anti-Wick Quantum States

Gosson

2020

Landscapes of Time-Frequency Analysis

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Convolutions for localization operators

Cited by 44 publications

References 29 publications

On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators

On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators

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

Generalized Anti-Wick Quantum States

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