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Gabor representations of evolution operators


Authors: Elena Cordero, Fabio Nicola and Luigi Rodino
Journal: Trans. Amer. Math. Soc. 367 (2015), 7639-7663
MSC (2010): Primary 35S05, 42C15
DOI: https://doi.org/10.1090/S0002-9947-2015-06302-8
Published electronically: March 2, 2015
MathSciNet review: 3391896
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Abstract: We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultra-analytic regularity. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schrödinger-type propagators, reveal to be an even more efficient tool for representing solutions to a wide class of evolution operators with constant coefficients, including weakly hyperbolic and parabolic-type operators. Besides the class of operators, the main novelty of the paper is the proof of super-exponential (as opposed to super-polynomial) off-diagonal decay for the Gabor matrix representation.


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Elena Cordero
Affiliation: Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
Email: elena.cordero@unito.it

Fabio Nicola
Affiliation: Dipartimento di Scienze Matematiche, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email: fabio.nicola@polito.it

Luigi Rodino
Affiliation: Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
Email: luigi.rodino@unito.it

DOI: https://doi.org/10.1090/S0002-9947-2015-06302-8
Keywords: Pseudodifferential operators, Gelfand-Shilov spaces, short-time Fourier transform, Gabor frames, sparse representations, hyperbolic equations, parabolic equations
Received by editor(s): July 23, 2013
Published electronically: March 2, 2015
Article copyright: © Copyright 2015 American Mathematical Society