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

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.