Localization of multidimensional Wigner distributions

Lieb, Élliott H.; Ostrover, Yaron

doi:10.1063/1.3486068

Cited by 9 publications

(10 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If we wish to find the functions ψ ∈ L 2 (R d ) whose Wigner distributions is maximally concentrated in a domain Ω ⊂ R 2d , proposition 8.2 5 reduces the problem to finding the eigenfunctions of the Weyl transform L χΩ . This insight was first formulated in Flandrin's paper [28], and extensions of his results include [59] and [52].…”

Section: Equation (10) States the Familiar Relationmentioning

confidence: 88%

“…The right hand side of this equation may be interpreted as a measure of the concentration of the energy of ψ in the region Ω of the time-frequency plane, and leads to a natural localization problem for Cohen's class [28,52,59,60] : for a Cohen's class distribution Q and a measurable Ω ⊂ R 2d . Find the signal ψ ∈ L 2 (R d ) with ψ L 2 = 1 that maximizes Ω Q(ψ)(z) dz.…”

Section: Equation (10) States the Familiar Relationmentioning

confidence: 99%

See 1 more Smart Citation

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

Luef

Skrettingland

2019

J Fourier Anal Appl

View full text Add to dashboard Cite

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 ...

show abstract

Section: Equation (10) States the Familiar Relationmentioning

confidence: 88%

Section: Equation (10) States the Familiar Relationmentioning

confidence: 99%

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

Luef

Skrettingland

2019

J Fourier Anal Appl

View full text Add to dashboard Cite

show abstract

“…Remark 9 Before we conclude this section, we remark that the optimization problem considered in this paper is intimately related with the so-called localization or Toeplitz operators in time-frequency analysis [16,18,42,45,46] and quantum mechanics [11,34]. In [11] the authors addressed the optimization problems…”

Section: A Particular Case In D =mentioning

confidence: 99%

“…In [11,18,34] the authors proved that Gaussians maximize (115), when D is a disk, a poly-disk or a ball.…”

Section: A Particular Case In D =mentioning

confidence: 99%

See 1 more Smart Citation

What is the Wigner Function Closest to a Given Square Integrable Function?

Ben-Benjamin¹,

Cohen²,

Dias³

et al. 2018

SIAM J. Math. Anal.

View full text Add to dashboard Cite

We consider an arbitrary square integrable function F on the phase space and look for the Wigner function closest to it with respect to the L 2 norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert-Schmidt operator with Weyl symbol F . We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a bona fide Wigner function. This is our prime motivation for our optimization problem. As a by-product of our results we are able to estimate the eigenvalues and Schatten norms of certain Schatten-class operators. The techniques presented here may be potentially interesting for estimating eigenvalues of localization operators in time-frequency analysis and quantum mechanics.

show abstract

On Some Singular Integrals Linked to Solvability and Signal Theory

Lerner

2024

Latin American Mathematics Series

View full text Add to dashboard Cite

Localization of multidimensional Wigner distributions

Cited by 9 publications

References 17 publications

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

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

What is the Wigner Function Closest to a Given Square Integrable Function?

On Some Singular Integrals Linked to Solvability and Signal Theory

Contact Info

Product

Resources

About