Gonchar-Stahl's $\rho^2$-theorem and associated directions in the theory of rational approximation of analytic functions

TL;DR

This paper discusses Gonchar-Stahl's $ ho^2$-theorem, which describes the convergence rate of best rational approximations of analytic functions, highlighting its significance and foundational contributions in rational approximation theory.

Contribution

It provides an overview of the $ ho^2$-theorem, its modifications, and related methods, emphasizing its role in the theory of rational approximation and complex analysis.

Findings

Characterizes convergence rates of rational approximations

Summarizes key contributions by Gonchar and Stahl

Outlines methods and generalizations of the theorem

Abstract

Gonchar-Stahl's $\rho^2$-theorem characterizes the rate of convergence of best uniform (Chebyshev) rational approximations (with free poles) for one basic class of analytic functions. The theorem itself, its modifications and generalizations, methods involved in the proof and other related details constitute an important subfield in the theory of rational approximations of analytic functions and complex analysis. The paper briefly outlines essentials of the subfield. Fundamental contributions by A. A. Gonchar and H. Stahl are in the center of the exposition.