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

摘要

We study the asymptotics of singular values and singular functions of aFinite Hilbert transform (FHT), which is defined on several intervals.Transforms of this kind arise in the study of the interior problem oftomography. We suggest a novel approach based on the technique of the matrixRiemann-Hilbert problem and the steepest descent method of Deift-Zhou. Weobtain 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 ordersolution 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 bystudying the intersections of the locus of zeroes of a certain Theta functionwith a straight line. This line can be calculated explicitly, and it depends onthe geometry of the intervals that define the FHT. The leading orderasymptotics of the singular functions and singular values are explicitlyexpressed in terms of the Riemann Theta functions and of the period matrix ofthe corresponding normalized differentials, respectively. We also obtain theerror estimates for our asymptotic results.