Exponential instability in the fractional Calder\'on problem

TL;DR

This paper demonstrates the exponential instability of the fractional Calderón problem, establishing the optimality of existing stability estimates, and explores related approximation and transform properties in one dimension.

Contribution

It proves exponential instability for the fractional Calderón problem, confirming the optimality of known logarithmic stability estimates and relating the problem to classical operators.

Findings

Exponential instability is established for the fractional Calderón problem.

The optimality of the logarithmic stability estimate from prior work is confirmed.

A close relation between the fractional Calderón problem and the truncated Hilbert transform in 1D is shown.

Abstract

In this note we prove the exponential instability of the fractional Calder\'on problem and thus prove the optimality of the logarithmic stability estimate from \cite{RS17}. In order to infer this result, we follow the strategy introduced by Mandache in \cite{M01} for the standard Calder\'on problem. Here we exploit a close relation between the fractional Calder\'on problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in \cite{RS17}. Finally, in one dimension, we show a close relation between the fractional Calder\'on problem and the truncated Hilbert transform.