Cross-ambiguity and Wigner Functions

  • Maurice A. de Gosson
Part of the Pseudo-Differential Operators book series (PDO, volume 7)


The Heisenberg–Weyl and Grossmann–Royer operators allow us to define in a particular simple way two classical objects from symplectic harmonic analysis, namely the cross-ambiguity and Wigner functions, which are symplectic Fourier transforms of each other. Wigner introduced the eponymic distribution in 1932 as a substitute for a phase space probability density, but he did that in an ad hoc way, a kind of “lucky guess” one could say. It has since then been realized that the Wigner distribution (and its companion, the cross-ambiguity function) have a very natural meaning in Weyl calculus, and that they can be simply defined in terms of the Grossmann–Royer and Heisenberg–Weyl operators of last chapter.


Wigner Function Inversion Formula Modulation Space Wigner Distribution Ambiguity Function 
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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Faculty of Mathematics, NuHAGUniversity of ViennaViennaAustria

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