Finite Hilbert transform with incomplete data: null-space and singular values

Abstract

Using the Gelfand–Graev formula, the interior problem of tomography reduces to the inversion of the finite Hilbert transform (FHT) from incomplete data. In this paper, we study several aspects of inverting the FHT when the data are incomplete. Using the Cauchy transform and an approach based on the Riemann–Hilbert problem, we derive a differential operator that commutes with the FHT. Our second result is the characterization of the null-space of the FHT in the case of incomplete data. Also, we derive the asymptotics of the singular values of the FHT in three different cases of incomplete data.