Time-Frequency Analysis of Fourier Integral Operators
Provides a new theoretical tool for analyzing FIOs in time-frequency analysis, but the results are incremental as they recapture known boundedness results.
The paper uses Gabor frames to efficiently represent Fourier Integral Operators (FIOs), showing that the matrix representation is well-organized. This leads to boundedness results on modulation spaces, recapturing known results for pseudo-differential operators, unimodular Fourier multipliers, and metaplectic operators.
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, 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 unimodular Fourier multipliers and metaplectic operators.