Interpolation characteristics of maximal polynomial approximants to rational functions
Volume 123 / 2019
Annales Polonici Mathematici 123 (2019), 155-169
MSC: 30E10, 41A05, 41A10.
DOI: 10.4064/ap180803-4-4
Published online: 13 August 2019
Abstract
Let $E$ be a compact set in $\mathbb C $ with connected regular complement and let $p_n$, $n \in \mathbb N $, be a sequence of polynomials which converge maximally to a fixed rational function $f$ on $E$. Then $p_n$ has $n + o(n)$ interpolation points to $f$ in $\mathbb C $ and the normalized counting measure $\nu _n$ of these interpolation points (resp. its balayage measure $\widehat {\nu }_n$ onto the boundary of $E$) converges to the equilibrium measure of $E$ as $n \rightarrow \infty $. Furthermore, we prove a complete characterization of maximal convergence by interpolation.
