Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature

Abstract

For a one-parameter family of simple metrics of constant curvature ($4κ$ for $κ\in (-1,1)$) on the unit disk $M$, we first make explicit the Pestov-Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also translates into moment conditions {\it à la} Helgason-Ludwig or Gel'fand-Graev. We then derive an explicit Singular Value Decomposition for the geodesic X-ray transform. Computations dictate a specific choice of weighted $L^2-L^2$ setting which is equivalent to the $L^2(M, dVol_κ)\to L^2(\partial_+ SM, dΣ^2)$ one for any $κ\in (-1,1)$.