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

For a one-parameter family of simple metrics of constant curvature ( 4\kappa for \kappa\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 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, \operatorname{dVol}_\kappa)\to L^2(\partial_+SM,d\Sigma^2) one for any \kappa\in (-1,1) .