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We use time-frequency methods for the study of Fourier Integral operators (FIOs). In this paper we shall show that Gabor frames provide very efficient representations for a large class of FIOs. Indeed, similarly to the case of shearlets and curvelets frames [6,27], 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 pseudodifferential operators with symbols in M ∞,1 [25], for some unimodular Fourier multipliers [2] and metaplectic operators [10,23].2000 Mathematics Subject Classification. 35S30,47G30,42C15.

We construct a one-parameter family of algebras F IO(Ξ, s), s ≥ 0, consisting of Fourier integral operators. We derive boundedness results, composition rules, and the spectral invariance of the operators in F IO(Ξ, s). The operator algebra is defined by the decay properties of an associated Gabor matrix around the graph of the canonical transformation.2000 Mathematics Subject Classification. 35S30, 47G30, 42C15.