Distribution of points of interpolation and of zeros of exact maximally convergent multipoint Pad\'e approximants

TL;DR

This paper investigates the distribution of interpolation points and zeros of maximally convergent multipoint Padé approximants for functions holomorphic on a compact set, establishing conditions for uniform distribution and zero behavior.

Contribution

It proves that maximally convergent multipoint Padé approximants have interpolation points uniformly distributed on the boundary and describes zeros' behavior under certain density conditions.

Findings

Interpolation points are uniformly distributed on the boundary with respect to measure μ.

Zeros of the approximants exhibit specific behavior under dense convergence conditions.

Maximal domain of meromorphic continuation is characterized by the convergence properties.

Abstract

Given a regular compact set $E$ in the complex plane, a unit measure $\mu$ supported by $\partial E,$ a triangular point set $\beta := \{\{\beta_{n,k}\}_{k=1}^n\}_{n=1}^{\infty},\beta\subset \partial E$ and a function $f$, holomorphic on $E$, let $\pi_{n,m}^{\beta,f}$ be the associated multipoint $\beta-$ Pad\'e approximant of order $(n,m)$. We show that if the sequence $\pi_{n,m}^{\beta,f}, n\in\Lambda, m-$ fixed, converges exact maximally to $f$, as $n\to\infty,n\in\Lambda$ inside the maximal domain of $m-$ meromorphic continuability of $f$ relatively to the measure $\mu,$ then the points $\beta_{n,k}$ are uniformly distributed on $\partial E$ with respect to the measure $\mu$ as $ n\in\Lambda$. Furthermore, a result about the zeros behavior of the exact maximally convergent sequence $\Lambda$ is provided, under the condition that $\Lambda$ is "dense enough."