Quantifying the ill-conditioning of analytic continuation

TL;DR

This paper analyzes the ill-conditioning of analytic continuation, showing it is manageable in an annulus but exponentially severe in strips, explaining the success of some numerical methods and limitations in others.

Contribution

It quantifies the ill-conditioning of analytic continuation in different regions, providing theoretical insights into the effectiveness of numerical methods like Chebfun.

Findings

Ill-conditioning is moderate in an annulus due to Hadamard's theorem.

Severe exponential ill-conditioning occurs in strips or channels.

Classical methods lose digits faster than modern approaches.

Abstract

Analytic continuation is ill-posed, but becomes merely ill-conditioned (although with an infinite condition number) if it is known that the function in question is bounded in a given region of the complex plane. In an annulus, the Hadamard three-circles theorem implies that the ill-conditioning is not too severe, and we show how this explains the effectiveness of Chebfun and related numerical methods in evaluating analytic functions off the interval of definition. By contrast, we show that analytic continuation is far more ill-conditioned in a strip or a channel, with exponential loss of digits of accuracy at the rate $\exp(-\pi x/2)$ as one moves along. The classical Weierstrass chain-of-disks method loses digits at the faster rate $\exp(-e\kern .3pt x)$.