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 ($4\kappa$ 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 $\grave{a}$ $la$ 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 $L^2(M,{\mathrm{dVol}}_{\kappa })\to L^2({\partial }_{+}SM,d{\Sigma }^2)$ one for any $\kappa \in (-1,1)$.