# Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature

## TL;DR

This paper provides explicit range characterizations and a singular value decomposition for the geodesic X-ray transform on disks with constant curvature, enhancing understanding of its mathematical structure and computational aspects.

## Contribution

It explicitly characterizes the range and co-kernel of the geodesic X-ray transform on constant curvature disks and derives its singular value decomposition.

## Key findings

- Explicit basis for the transform's range and co-kernel.
- Moment conditions analogous to classical integral geometry.
- A specific weighted $L^2$ setting for computations.

## 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 {\it \`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 $L^2(M, dVol_\kappa)\to L^2(\partial_+ SM, d\Sigma^2)$ one for any $\kappa\in (-1,1)$.