n-th Root Optimal Rational Approximants to Functions with Polar Singular Set

TL;DR

This paper studies the asymptotic behavior of optimal rational approximants to functions with polar singularities, focusing on their pole distribution and convergence properties using potential theory on Riemann surfaces.

Contribution

It introduces a framework for analyzing n-th root optimal rational approximants to functions with polar singularities, extending to meromorphic approximants and employing potential theory.

Findings

Weak* convergence of pole measures established

Convergence in capacity demonstrated

Behavior characterized via potential theory on Riemann surfaces

Abstract

Let $ D $ be a bounded Jordan domain and $ A $ be its complement on the Riemann sphere. We investigate the $ n $-th root asymptotic behavior in $ D $ of best rational approximants, in the uniform norm on $ A $, to functions holomorphic on $ A $ having a multi-valued continuation to quasi every point of $ D $ with finitely many branches. More precisely, we study weak$^*$ convergence of the normalized counting measures of the poles of such approximants as well as their convergence in capacity. We place best rational approximants into a larger class of $ n $-th root optimal meromorphic approximants, whose behavior we investigate using potential-theory on certain compact bordered Riemann surfaces.