Convergence of two-point Pad\'e approximants to piecewise holomorphic functions

TL;DR

This paper investigates the uniform convergence of two-point Padé approximants for piecewise holomorphic functions with branch points, focusing on cases where the convergence set separates the plane, extending Buslaev's capacity convergence results.

Contribution

It analyzes the uniform convergence behavior of two-point Padé approximants on separating sets, providing new insights into their convergence properties for multi-valued functions.

Findings

Convergence occurs on certain separating sets for the approximants.

The set F can or cannot separate the plane, affecting convergence.

The study extends capacity convergence results to uniform convergence cases.

Abstract

Let $ f_0 $ and $ f_\infty $ be formal power series at the origin and infinity, and $ P_n/Q_n $, with $ \mathrm{deg}(P_n),\mathrm{deg}(Q_n)\leq n $, be a rational function that simultaneously interpolates $ f_0 $ at the origin with order $ n $ and $ f_\infty $ at infinity with order $ n+1 $. When germs $ f_0,f_\infty $ represent multi-valued functions with finitely many branch points, it was shown by Buslaev that there exists a unique compact set $ F $ in the complement of which the approximants converge in capacity to the approximated functions. The set $ F $ might or might not separate the plane. We study uniform convergence of the approximants for the geometrically simplest sets $ F $ that do separate the plane.