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 (4? for ??(?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 � 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 L2?L2 setting which is equivalent to the L2(M,dVol?)?L2(?+SM,d?2) one for any ??(?1,1).