On characterization of the range and inversion formulas for the geodesic X-ray transform

We describe a relation between the scattering relation, the Hilbert transform in frequency space, and the geodesic ray transform for simple, two-dimensional compact Riemannian manifolds with boundary. We use this relation to give a characterization of the range of the geodesic X-ray transform acting on scalar functions and vector fields in terms of the scattering relation which involves only measurements at the boundary. We also give inversion formulas for the geodesic X-ray transform acting on scalar functions and vector fields.