Time-Frequency Analysis of Fourier Integral Operators

TL;DR

This paper demonstrates that Gabor frames provide efficient matrix representations for Fourier Integral Operators, enabling analysis of their boundedness on modulation spaces and recapturing known results for related operators.

Contribution

It introduces the use of Gabor frames for representing FIOs, facilitating the study of their boundedness on modulation spaces with a well-organized matrix structure.

Findings

Gabor frames yield efficient representations of FIOs.

The matrix representation helps analyze boundedness on modulation spaces.

Recovers known boundedness results for pseudo-differential operators and Fourier multipliers.

Abstract

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.