On accumulated spectrograms

Abreu, Luís Daniel; Gröchenig, Karlheinz; Romero, José Luis

doi:10.1090/tran/6517

Cited by 71 publications

(81 citation statements)

References 32 publications

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“…Numerical experiments show that when N is close to the critical value ≈ |Ω|, the sum of the N spectrograms that are most concentrated on Ω almost exhaust the domain Ω; i.e., the accumulated spectrogram looks approximately like 1 Ω . The following result from [2] provides a rigorous formulation of this observation.…”

Section: (When the Underlying Window G Is Clear From The Context We mentioning

confidence: 96%

“…In addition, Theorem 1.1 does not offer numerical stability: from the information that the eigenmodes of a time-frequency localization operator H Ω look approximately like Hermite functions, one cannot conclude that the localization domain Ω is approximately a disk. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A more feasible approach to the approximate recovery of the localization domain from partial spectral information has been developed in [2], based on the concept of accumulated spectrogram. It provides a method for approximating the symbol of a localization operator from the intensities of the time-frequency representations of its eigenfunctions.…”

Section: 3mentioning

confidence: 99%

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Sharp rates of convergence for accumulated spectrograms

2017

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Abstract. We investigate an inverse problem in time-frequency localization: the approximation of the symbol of a time-frequency localization operator from partial spectral information by the method of accumulated spectrograms (the sum of the spectrograms corresponding to large eigenvalues). We derive a sharp bound for the rate of convergence of the accumulated spectrogram, improving on recent results.

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Section: (When the Underlying Window G Is Clear From The Context We mentioning

confidence: 96%

Section: 3mentioning

confidence: 99%

Sharp rates of convergence for accumulated spectrograms

2017

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“…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%

“…This is a prerequisite for generalizing the results [2,3]. A key fact is that the approach in [2,3] is only feasible in the case of rank-one operators. For a general density operator one has to develop a different strategy.…”

Section: Introductionmentioning

confidence: 99%

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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“…Remark 1: The above construction applies to several settings where properties similar to Theorem A are available, as in [3] and [9], [4].…”

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confidence: 99%

Pseudo prolate spheroidal functions

Abreu

Pereira

2015

2015 International Conference on Sampling Theory and Applications (SampTA)

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Abstract-Let. We have recently revisited this problem with a new approach: instead of counting the number of eigenfunctions with eigenvalue close to one, we count the maximum number of orthogonal ǫ-pseudoeigenfunctions with ǫ-pseudoeigenvalue one. Precisely, we count how many orthogonal functions have a maximum of energy ǫ outside the domain T × Ω, in the sense that PT,Ωf − f 2 ≤ ǫ.We have recently discovered that the sharp asymptotic number is. The proof involves an explicit construction of the pseudoeigenfunctions of PT,Ω. When T and Ω are intervals we call them pseudo prolate spheroidal functions. In this paper we explain how they are constructed.

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On accumulated spectrograms

Cited by 71 publications

References 32 publications

Sharp rates of convergence for accumulated spectrograms

Sharp rates of convergence for accumulated spectrograms

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

Pseudo prolate spheroidal functions

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