Time-frequency representations of Wigner type and pseudo-differential operators

Boggiatto, P.; De Donno, G.; Oliaro, A.

doi:10.1090/S0002-9947-10-05089-0

Time-frequency representations of Wigner type and pseudo-differential operators
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by P. Boggiatto, G. De Donno and A. Oliaro PDF

Trans. Amer. Math. Soc. 362 (2010), 4955-4981 Request permission

Abstract:

We introduce a $\tau$-dependent Wigner representation, $\operatorname {Wig}_\tau$, $\tau \in [0,1]$, which permits us to define a general theory connecting time-frequency representations on one side and pseudo-differential operators on the other. The scheme includes various types of time-frequency representations, among the others the classical Wigner and Rihaczek representations and the most common classes of pseudo-differential operators. We show further that the integral over $\tau$ of $\operatorname {Wig}_\tau$ yields a new representation $Q$ possessing features in signal analysis which considerably improve those of the Wigner representation, especially for what concerns the so-called “ghost frequencies”. The relations of all these representations with respect to the generalized spectrogram and the Cohen class are then studied. Furthermore, a characterization of the $L^p$-boundedness of both $\tau$-pseudo-differential operators and $\tau$-Wigner representations are obtained.

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