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 ∈ (−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 cokernel. 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, dV ol κ ) → L 2 (∂ + SM, dΣ 2 ) one for any κ ∈ (−1, 1).