Weighted Extremal Domains and Best Rational Approximation

TL;DR

This paper characterizes the asymptotic distribution of poles of best rational approximants to certain holomorphic functions, linking potential theory and approximation error estimates in complex analysis.

Contribution

It introduces a new potential-theoretic framework for understanding extremal domains and pole distribution in rational approximation of functions with branch points.

Findings

Poles of best rational approximants distribute according to the equilibrium measure.

Provides n-th root asymptotics of the approximation error.

Extends estimates to approximation on general Jordan curves.

Abstract

Let f be holomorphically continuable over the complex plane except for finitely many branch points contained in the unit disk. We prove that best rational approximants to f of degree n, in the L^2-sense on the unit circle, have poles that asymptotically distribute according to the equilibrium measure on the compact set outside of which f is single-valued and which has minimal Green capacity in the disk among all such sets. This provides us with n-th root asymptotics of the approximation error. By conformal mapping, we deduce further estimates in approximation by rational or meromorphic functions to f in the L^2-sense on more general Jordan curves encompassing the branch points. The key to these approximation-theoretic results is a characterization of extremal domains of holomorphy for f in the sense of a weighted logarithmic potential, which is the technical core of the paper.