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

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."