Singular Value Decomposition of a Finite Hilbert Transform Defined on Several Intervals and the Interior Problem of Tomography: The Riemann‐Hilbert Problem Approach

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 RiemannHilbert problem and the steepest descent method of Deift-Zhou. We obtain a family of matrix RHPs depending on the spectral parameter λ and show that the singular values of the FHT coincide with the values of λ for which the RHP is not solvable. Expressing the leading order solution as λ → 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.

On Sobolev instability of the interior problem of tomography

“…In [BKT13] it is shown that the asymptotic approximations to the exact singular functions f n are valid uniformly on compact subsets of the interior of I i as n → ∞. In Section 4 we show that these approximations are also valid in the L 2 (I i ) sense as well.…”

“…We do not consider the other set of singular functions that are defined on I e , since they are not needed for the analytic continuation of ψ. The main idea of the approach in [BKT13] is to reduce the SVD problem (1.9) to a matrix Riemann-Hilbert problem (RHP), which, in turn, is asymptotically reduced to a simpler RHP. That simpler (model) RHP has an explicit solution, which can be expressed in terms of the Riemann Theta function.…”

“…That simpler (model) RHP has an explicit solution, which can be expressed in terms of the Riemann Theta function. A brief review of the reduction to the model RHP and certain related results from [BKT13] are contained in Appendix A. Some technical lemmas related to the approximation of singular functions on [a 1 , a 2g+2 ] \ I and on I i that are needed in Sections 3 and 4 are proven in Appendix B.…”

“…For convenience, most of the statements below are provided with direct references (in square brackets) to the corresponding results of [BKT13].…”

“…Our main tool is the singular value decomposition of the operator H −1 e that arises when the interior problem is reduced to a problem of inverting the Hilbert transform from incomplete data. The asymptotics of the singular values and singular functions of H −1 e , the latter being valid uniformly on compact subsets of the interior of Ii, was obtained in [BKT13]. Using these asymptotics we can accurately measure the degree of ill-posedness of the analytic continuation as a function of the target interval J.…”