Range characterizations and singular value decomposition of the geodesic X-ray transform on disks of constant curvature (Q2078204)

Summary: 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 \(\kappa\in (-1,1)\).