Radon transform, spherical means, and an inverse problem for the wave equation

Abstract

Consider the linear map which sends the initial data to the trace, on the light cone with vertex at 0, of the solution of the initial value problem for the wave equation. We show this map is an isometry, construct its inverse, and give a partial characterization of its range. Our results are better in odd space dimensions than in the even dimensional case. Solutions of the wave equation can be expressed in terms of the spherical averages of the initial data and these spherical averages can be related to the Radon transform of a related function. We obtain our results exploiting this relationship and the isometry, the inversion formula and the range characterization of the Radon transform.