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

Given a regular compact set $E$ in the complex plane, a unit measure $\mu$ supported by $\mathrm{\partial }E,$ 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 ${\pi }_{n,m}^{\beta ,f},n\in \mathrm{\Lambda },m-$ fixed, converges exact maximally to $f$, as $n\to \mathrm{\infty },n\in \mathrm{\Lambda }$ 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 $\mathrm{\partial }E$ with respect to the measure $\mu$ as $n\in \mathrm{\Lambda }$. Furthermore, a result about the zeros behavior of the exact maximally convergent sequence $\mathrm{\Lambda }$ is provided, under the condition that $\mathrm{\Lambda }$ is “dense enough.”