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Linear Perturbations of the Wigner Transform and the Weyl Quantization

  • Dominik Bayer
  • Elena CorderoEmail author
  • Karlheinz Gröchenig
  • S. Ivan Trapasso
Chapter
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Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal’s formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen’s class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.

Keywords

Time-frequency analysis Wigner distribution Cohen’s class Modulation space Pseudodifferential operator Quantization 

2010 Mathematics Subject Classification

42A38 42B35 46F10 46F12 81S30 

Notes

Acknowledgements

E. Cordero and S. I. Trapasso are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). K. Gröchenig acknowledges support from the Austrian Science Fund FWF, project P31887-N32.

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Dominik Bayer
    • 1
  • Elena Cordero
    • 2
    Email author
  • Karlheinz Gröchenig
    • 3
  • S. Ivan Trapasso
    • 4
  1. 1.Universität der Bundeswehr MünchenMünchenGermany
  2. 2.Dipartimento di MatematicaUniversità di TorinoTorinoItaly
  3. 3.Faculty of MathematicsUniversity of ViennaViennaAustria
  4. 4.Dipartimento di Scienze Matematiche “G. L. Lagrange”Politecnico di TorinoTorinoItaly

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