On Sobolev instability of the interior problem of tomography

Abstract

In this paper we continue investigation of the interior problem of tomography that was started in \cite{BKT2}. As is known, solving the interior problem {with prior data specified on a finite collection of intervals $I_i$} is equivalent to analytic continuation of a function from $I_i$ to an open set ${\bf J}$. In the paper we prove that this analytic continuation can be obtained with the help of a simple explicit formula, which involves summation of a series. Our second result is that the operator of analytic continuation is not stable for any pair of Sobolev spaces regardless of how close the set ${\bf J}$ is to $I_i$. Our main tool is the singular value decomposition of the operator $\mathcal H^{-1}_e$ 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 $\mathcal H^{-1}_e$, the latter being valid uniformly on compact subsets {of the interior of $I_i$}, was obtained in \cite{BKT2}. {Using these asymptotics we can accurately measure the degree of ill-posedness of the analytic continuation as a function of the target interval ${\bf J}$.} Our {last} result is the convergence of the asymptotic approximation of the singular functions {in the $L^2(I_i)$ sense}.