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, 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}, for some Fourier multipliers and metaplectic operators. Moreover, this paper provides the mathematical tools to numerically solving the Cauchy problem for Schrödinger equations using Gabor frames. Finally, similar arguments can be employed to study other classes of FIOs .