A Parametrix Construction for the Wave Equation with Low Regularity Coefficients Using a Frame of Gaussians

TL;DR

This paper develops a Gaussian frame for L^2 functions to construct a parametrix for the wave equation with low regularity coefficients, extending to Gaussian beams with higher regularity.

Contribution

It introduces a Gaussian frame construction for low regularity coefficients and demonstrates its use in building a wave equation parametrix, bridging to Gaussian beams in smoother cases.

Findings

Constructed a Gaussian frame for L^2 functions.

Built a parametrix for the wave equation using propagated frame functions.

Extended the approach to Gaussian beams with more regular coefficients.

Abstract

We construct a frame of complex Gaussians for the space of $L^2(\mathbb{R}^n)$ functions. When propagated along bicharacteristics for the wave equation, the frame can be used to build a parametrix with suitable error terms. When the coefficients of the wave equation have more regularity, propagated frame functions become Gaussian beams.