Gabor representations of evolution operators

TL;DR

This paper introduces advanced Gabor frame techniques for efficiently representing solutions to various evolution operators, demonstrating super-exponential decay in their matrix representations for better computational and analytical handling.

Contribution

It develops a novel super-exponential decay estimate for Gabor matrix representations of evolution operators with constant coefficients, enhancing sparse decomposition methods.

Findings

Gabor frames provide highly sparse representations of evolution operators.

Super-exponential off-diagonal decay is proven for Gabor matrix representations.

Applicable to hyperbolic and parabolic evolution equations.

Abstract

We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultra-analytic regularity. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schroedinger-type propagators, reveal to be an even more efficient tool for representing solutions to a wide class of evolution operators with constant coefficients, including weakly hyperbolic and parabolic-type operators. Besides the class of operators, the main novelty of the paper is the proof of super-exponential (as opposite to super-polynomial) off-diagonal decay for the Gabor matrix representation.