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