A Radon transform on spheres through the origin in $\textbf {R}^{n}$ and applications to the Darboux equation

Abstract:

On domain ${C^\infty } ({R^n})$ we invert the Radon transform that maps a function to its mean values on spheres containing the origin. Our inversion formula implies that if $f \in {C^\infty } ({R^n})$ and its transform is zero on spheres inside a disc centered at 0, then f is zero inside that disc. We give functions $f \notin {C^\infty } ({R^n})$ whose transforms are identically zero and we give a necessary condition for a function to be the transform of a rapidly decreasing function. We show that every entire function is the transform of a real analytic function. These results imply that smooth solutions to the classical Darboux equation are determined by the data on any characteristic cone with vertex on the initial surface; if the data is zero near the vertex then so is the solution. If the data is entire then a real analytic solution with that data exists.