Approximation of Fourier Integral Operators by Gabor multipliers

TL;DR

This paper demonstrates that Fourier integral operators can be effectively approximated by Gabor multipliers using time-frequency analysis, with precise error estimates for operators with smooth phases and symbols.

Contribution

It provides a rigorous framework for approximating Fourier integral operators with Gabor multipliers, extending the understanding of their matrix concentration near propagation curves.

Findings

Fourier integral operators are nearly diagonal in Gabor frame representation.

Explicit error bounds are derived for Gabor multiplier approximations.

The approach leverages time-frequency analysis techniques for operator approximation.

Abstract

A general principle says that the matrix of a Fourier integral operator with respect to wave packets is concentrated near the curve of propagation. We prove a precise version of this principle for Fourier integral operators with a smooth phase and a symbol in the Sjoestrand class and use Gabor frames as wave packets. The almost diagonalization of such Fourier integral operators suggests a specific approximation by (a sum of) elementary operators, namely modified Gabor multipliers. We derive error estimates for such approximations. The methods are taken from time-frequency analysis.