On Sobolev instability of the interior problem of tomography

As is known, solving the interior problem with prior data specified on a finite collection of intervals IiIi is equivalent to analytic continuation of a function from IiIi to an open set 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 J is to IiIi. Our main tool is the singular value decomposition of the operator He−1 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 He−1, the latter being valid uniformly on compact subsets of the interior of IiIi, was obtained in [5]. Using these asymptotics we can accurately measure the degree of ill-posedness of the analytic continuation as a function of the target interval J. Our last result is the convergence of the asymptotic approximation of the singular functions in the L2(Ii)L2(Ii) sense. We also present a preliminary numerical experiment, which illustrates how to use our results for reducing the instability of the analytic continuation by optimizing the position of the intervals with prior knowledge.