Singular value decomposition of a finite Hilbert transform defined on several intervals and the interior problem of tomography: the Riemann-Hilbert problem approach (Q2790836)

The authors study the asymptotics of singular values and singular functions of a finite Hilbert transform defined on several finite intervals. These types of transforms arise in the study of the interior problem of tomography. The authors suggest an approach based on the technique of the matrix Riemann-Hilbert problem and the steepest-decent method of Deift-Zhou. They obtain a family of matrix Riemann-Hilbert problems and prove that the asymptotics of the singular values can be obtained by studying the intersections of the locus of zeroes of a certain theta function with a straight line. The line depends on the geometry of the intervals that define the finite Hilbert transform. They also obtain the error estimates for the asymptotic results.