Recovering initial values from light cone traces of solutions of the wave equation

TL;DR

This paper addresses the problem of reconstructing initial data for the wave equation from light cone traces, establishing an isometric mapping in odd dimensions and linking it to spherical means and Radon transform inversion.

Contribution

It introduces an isometric map for initial value recovery from light cone traces in odd dimensions and characterizes its range and inverse, connecting to spherical means and Radon transform.

Findings

The map from initial value to light cone traces is an isometry in odd dimensions.

The range of this map is characterized explicitly.

An explicit inverse of the map is constructed.

Abstract

We consider the problem of recovering the initial value, from the trace on the light cone, of the solution of an initial value problem for the wave equation. When the space is odd dimensional, we show that the map from the initial value to the traces of the (even or odd in time) solutions on the light cone is an isometry and we characterize the range of this map and construct its inverse. We do this by relating the problem to the recovery of a function from its spherical means over all spheres through the origin, which in turn is related to the Radon transform inversion via the inversion map on R^n.