# American Institute of Mathematical Sciences

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

## Time-frequency analysis of fourier integral operators

 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].
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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