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

TL;DR

This paper develops a Riemann-Hilbert problem approach to analyze the asymptotics of singular values and functions of a finite Hilbert transform on multiple intervals, relevant for tomography's interior problem.

Contribution

It introduces a novel Riemann-Hilbert problem method to derive explicit asymptotics of singular values/functions for the finite Hilbert transform on several intervals.

Findings

Asymptotic formulas for singular values in terms of Riemann Theta functions.

Explicit characterization of singular functions using Riemann Theta functions.

Error estimates for the asymptotic approximations.

Abstract

We study the asymptotics of singular values and singular functions of a Finite Hilbert transform (FHT), which is defined on several intervals. Transforms of this kind arise in the study of the interior problem of tomography. We suggest a novel approach based on the technique of the matrix Riemann-Hilbert problem and the steepest descent method of Deift-Zhou. We obtain a family of matrix RHPs depending on the spectral parameter $\lambda$ and show that the singular values of the FHT coincide with the values of $\lambda$ for which the RHP is not solvable. Expressing the leading order solution as $\lambda\to 0$ of the RHP in terms of the Riemann Theta functions, we 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. This line can be calculated explicitly, and it depends on the geometry of the intervals that define the FHT. The leading order asymptotics of the singular functions and singular values are explicitly expressed in terms of the Riemann Theta functions and of the period matrix of the corresponding normalized differentials, respectively. We also obtain the error estimates for our asymptotic results.