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 constant curvature ($4\\kappa$ for $\\kappa\\in (-1,1)$) on the unit disk $M$, we first make explicit Pestov–Uhlmann range characterization geodesic X-ray transform, by constructing basis functions making up its and co-kernel. Such also translates into moment conditions $à$ $la$ Helgason–Ludwig or Gel'fand–Graev. We then derive an Singular Value Decomposition transform. Computations dictate specific choice weighted $L^2-L^2$ setting which is equivalent to $L^2(M, \\operatorname{dVol}\\kappa)\\to L^2(\\partial+SM,d\\Sigma^2)$ one any (-1,1)$.