Range conditions for a spherical mean transform and global extendibility of solutions of the darboux equation

Abstract

We describe the range of the spherical Radon transform which evaluates integrals of a function in IRn over all spheres centered on a given sphere. Such a transform attracts much attention due to its applications in approximation theory and (thermo- and photoacoustic) tomography. Range descriptions for this transform have been obtained recently. They include two types of conditions: an orthogonality condition and, for even n, a moment condition. It was later discovered that, in all dimensions, the moment condition follows from the orthogonality condition (and can therefore can be dropped). In terms of the Darboux equation, which describes spherical means, this indirectly implies that solutions of certain boundary value problems in a domain extend automatically outside of the domain. In this article, we present a direct proof of this global extendibility phenomenon for the Darboux equation. Correspondingly, we deliver an alternative proof of the range characterization theorem.