Distribution of points of interpolation and of zeros of exact maximally convergent multipoint Padé 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 $\beta -$ Padé 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."