On the optimal analytic continuation from discrete data

TL;DR

This paper derives optimal bounds for the problem of analytic continuation from discrete data within a reproducing kernel Hilbert space, characterizing the maximal possible error and its asymptotic behavior.

Contribution

It provides the first sharp bounds on analytic continuation errors and characterizes the extremal functions using operator resolvents.

Findings

Established optimal bounds on analytic continuation error.

Described the asymptotic behavior of the error in terms of data precision.

Characterized extremal functions via resolvent operators.

Abstract

We consider analytic functions from a reproducing kernel Hilbert space. Given that such a function is of order $\epsilon$ on a set of discrete data points, relative to its global size, we ask how large can it be at a fixed point outside of the data set. We obtain optimal bounds on this error of analytic continuation and describe its asymptotic behavior in $\epsilon$. We also describe the maximizer function attaining the optimal error in terms of the resolvent of a positive semidefinite, self-adjoint and finite rank operator.