Minkowski curvelets and wave equations

TL;DR

This paper introduces Minkowski curvelets, a new wavelet framework tailored for wave equations, exhibiting rapid decay properties and non-intersection with the light-cone, enhancing analysis of the Klein-Gordon operator.

Contribution

It develops Minkowski curvelets, a novel anisotropic wavelet basis for wave equations, with proven decay properties and specific Fourier support characteristics.

Findings

Green kernel matrix shows nearly exponential off-diagonal decay in Minkowski curvelet basis.

Minkowski curvelets are strongly anisotropic and avoid the light-cone in Fourier space.

The basis improves understanding of wave propagation and decay in Minkowski space.

Abstract

We define a new type of wavelet frame adapted to the study of wave equations, that we call Minkowski curvelets, by reference to the curvelets introduced by Cand\`es, Demanet and Donoho. These space-time, strongly anisotropic, directional wavelets have a Fourier support which does not intersect the light-cone; their maximal size is proportional to the inverse of the distance to the light-cone. We show that the matrix of the Green kernel of the Klein-Gordon operator on Minkowski space-time has a nearly exponential off-diagonal decay in this basis.