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January  2010, 9(1): 1-21. doi: 10.3934/cpaa.2010.9.1

Time-frequency analysis of fourier integral operators

Elena Cordero 1, , Fabio Nicola 2, and  Luigi Rodino 1,

1. 

Department of Mathematics, University of Torino, via Carlo Alberto 10, 10123 Torino, Italy, Italy
2. 

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received  January 2009 Revised  June 2009 Published  October 2009

Time-frequency methods are used to study a class of Fourier Integral Operators (FIOs) whose representation using Gabor frames is proved to be very efficient. Indeed, similarly to the case of shearlets and curvelets frames [10, 35], the matrix representation of a Fourier Integral Operator with respect to a Gabor frame is well-organized. This is used as a powerful tool to study the boundedness of FIOs on modulation spaces. As special cases, we recapture boundedness results on modulation spaces for pseudo-differential operators with symbols in $M^{\infty, 1}$ [33], for some Fourier multipliers [6] and metaplectic operators [14, 31]. Moreover, this paper provides the mathematical tools to numerically solving the Cauchy problem for Schr¨odinger equations using Gabor frames [17]. Finally, similar arguments can be employed to study other classes of FIOs [16].
Keywords: Fourier integral operators, modulation spaces, Gabor frames., short-time Fourier transform.
Mathematics Subject Classification: Primary: 35S30, 47G30, 42C1.
Citation: Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure & Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1
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