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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: P. Boggiatto, G. De Donno and A. Oliaro
Journal: Trans. Amer. Math. Soc. 362 (2010), 4955-4981
MSC (2010): Primary 47G30; Secondary 35S05, 42B10, 44A35, 47B38
Published electronically: April 21, 2010
MathSciNet review: 2645057
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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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Additional Information

P. Boggiatto
Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, I-10123 Torino, Italy
Email: paolo.boggiatto@unito.it

G. De Donno
Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, I-10123 Torino, Italy
Email: giuseppe.dedonno@unito.it

A. Oliaro
Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, I-10123 Torino, Italy
Email: alessandro.oliaro@unito.it

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05089-0
PII: S 0002-9947(10)05089-0
Received by editor(s): April 3, 2008
Received by editor(s) in revised form: April 17, 2009
Published electronically: April 21, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.