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The conditional Weyl transform and its generalization

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Abstract

The expectation value of the Weyl transform of a symbol with a state function equals the phase-space averaging of the symbol with the Wigner distribution. We define the conditional Weyl transform so that its expectation value equals the conditional average of the symbol taken with the Wigner distribution. Furthermore, we generalize to arbitrary operator correspondences and the generalized phase-space distributions.

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Notes

  1. We note that in different fields the symbol is called the c-function, where c stands for classical function. In time-frequency analysis it is called the ordinary function of time and frequency. The Weyl transform is often called the Weyl operator, the Weyl correspondence, the Weyl ordering rule, or the Weyl rule of association. All integrals go from –\(\infty \) to \(\infty \) unless otherwise noted.

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Authors and Affiliations

  1. Department of Physics, Hunter College of the City University of New York, 695 Park Ave, New York, NY, 10021, USA

    Leon Cohen

  2. Department of Bioengineering, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Patrick Loughlin

Authors
  1. Leon Cohen
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  2. Patrick Loughlin
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Correspondence to Leon Cohen.

Additional information

The research was supported by the Office of Naval Research; grant numbers N00014-09-1-0162 (LC) and N00014-10-1-0053 (PL)

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Cohen, L., Loughlin, P. The conditional Weyl transform and its generalization. J. Pseudo-Differ. Oper. Appl. 4, 1–12 (2013). https://doi.org/10.1007/s11868-012-0059-4

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  • DOI: https://doi.org/10.1007/s11868-012-0059-4

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