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Windowed-Wigner Representations in the Cohen Class and Uncertainty Principles

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Abstract

For representations in the Cohen class, specific Cohen kernels depending only on one half of the variables are showed to produce two types of representations which can in a natural way be associated with time and frequency windows. This leads to the definition of representations with no interference for signals whose time-frequency content is confined in specific zones. We prove the main properties of these representations in the context of the Cohen class. We study then uncertainty principles at first in connection with support compactness and then in the framework of a general concept of duality among representations.

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Notes

  1. The choice R=0.9 is connected with the “essential” support of \(\hat{f}(\omega)\), i.e., the set where \(\hat{f}(\omega)\) is not “too” small; here R plays the role of M in (1.4); see Figure 1.

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Authors and Affiliations

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

    Paolo Boggiatto, Evanthia Carypis & Alessandro Oliaro

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  1. Paolo Boggiatto
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  2. Evanthia Carypis
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  3. Alessandro Oliaro
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Correspondence to Paolo Boggiatto.

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Communicated by Loukas Grafakos.

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Boggiatto, P., Carypis, E. & Oliaro, A. Windowed-Wigner Representations in the Cohen Class and Uncertainty Principles. J Geom Anal 23, 1753–1779 (2013). https://doi.org/10.1007/s12220-012-9304-6

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  • DOI: https://doi.org/10.1007/s12220-012-9304-6

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