TIME-FREQUENCY ANALYSIS OF FOURIER INTEGRAL OPERATORS

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-infinity,M-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 Schrodinger equations using Gabor frames [17]. Finally, similar arguments can be employed to study other classes of FIOs [16].