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Time–Frequency Localization Operators: State of the Art

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Landscapes of Time-Frequency Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

We present localization operators via the short-time Fourier transform. For both modulation and ultra-modulation spaces framework, well-known results about boundedness and Schatten-von Neumann class are reported. Asymptotic eigenvalues’ distribution and decay and smoothness properties for L 2-eigenfunctions are exhibited. Eventually, we make a conjecture about smoothness of L 2-eigenfunctions for localization operators with Gelfand–Shilov windows and symbols in ultra-modulation spaces.

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Acknowledgements

I express my gratitude to Professor Elena Cordero for her time and valuable advice.

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, Torino, Italy

    Federico Bastianoni

Authors
  1. Federico Bastianoni
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Correspondence to Federico Bastianoni .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Turin, Torino, Italy

    Paolo Boggiatto

  2. Department of Mathematics, Ghent University, Ghent, Belgium

    Tommaso Bruno

  3. Department of Mathematics, University of Turin, Torino, Italy

    Elena Cordero

  4. Institute of Mathematics, University of Vienna, Wien, Austria

    Hans G. Feichtinger

  5. Department of Mathematical Sciences, Polytechnic University of Turin, Torino, Italy

    Fabio Nicola

  6. Department of Mathematics, University of Turin, Torino, Italy

    Alessandro Oliaro

  7. Department of Mathematical Sciences, Polytechnic University of Turin, Torino, Italy

    Anita Tabacco

  8. Department of Mathematical Sciences, Polytechnic University of Turin, Torino, Italy

    Maria Vallarino

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Bastianoni, F. (2020). Time–Frequency Localization Operators: State of the Art. In: Boggiatto, P., et al. Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-56005-8_4

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