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Discrete Analogs of Wigner Transforms and Weyl Transforms

  • Shahla Molahajloo
  • M. W. WongEmail author
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

We first introduce the discrete Fourier–Wigner transform and the discrete Wigner transform acting on functions in \(L^2({\mathbb Z})\). We prove that properties of the standard Wigner transform of functions in \(L^2({\mathbb R}^n)\) such as the Moyal identity, the inversion formula, time-frequency marginal conditions, and the resolution formula hold for the Wigner transforms of functions in \(L^2({\mathbb Z})\). Using the discrete Wigner transform, we define the discrete Weyl transform corresponding to a suitable symbol on \({\mathbb Z}\times {\mathbb S}^1\). We give a necessary and sufficient condition for the self-adjointness of the discrete Weyl transform. Moreover, we give a necessary and sufficient condition for a discrete Weyl transform to be a Hilbert–Schmidt operator. Then we show how we can reconstruct the symbol from its corresponding Weyl transform. We prove that the product of two Weyl transforms is again a Weyl transform and an explicit formula for the symbol of the product of two Weyl transforms is given. This result gives a necessary and sufficient condition for the Weyl transform to be in the trace class.

Keywords

Fourier–Wigner transform Wigner transform Weyl transform Moyal identity Time-frequency marginal conditions Wigner inversion formula Weyl inversion formula Kernels Hilbert–Schmidt operators Trace class operators Twisted convolution Weyl calculus 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsInstitute for Advanced Studies in Basic SciencesZanjanIran
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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