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

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Abstract

Recently we introduced mixed-state localization operators associated with 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 that is based on the Cohen class distribution associated with 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 short-time 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.

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Notes

  1. We always assume this is antilinear in the second coordinate, to be consistent with the inner product on \(L^2(\mathbb {R}^d)\).

  2. The alert reader will note that we use \( (\chi _\Omega \star S) \check{}=\check{\chi _\Omega }\star \check{S}\). See [41] for the simple proof.

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  1. Department of Mathematics, NTNU Norwegian University of Science and Technology, NO–7491, Trondheim, Norway

    Franz Luef & Eirik Skrettingland

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  1. Franz Luef
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Correspondence to Franz Luef.

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Communicated by Rémi Gribonval.

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Luef, F., Skrettingland, E. On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators. Constr Approx 52, 31–64 (2020). https://doi.org/10.1007/s00365-019-09465-2

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  • DOI: https://doi.org/10.1007/s00365-019-09465-2

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