Computational Methods and Function Theory 6 (2006), No. 1, 203--222
Copyright Heldermann Verlag 2006
Anders Gustafsson
addeladde@gmail.com , Department of Engineering, Physics and Mathematics, Mid Sweden University, 851\,70 Sundsvall, Sweden.
Let $D$ be a regular domain in $\overline{\mathbb{C}}$ with bounded boundary. For each $n\geq 1$, take points $A_n=\{a_{ni}\}_{i=0}^n$ in $D$ and points $B_n=\{b_{ni}\}_{i=1}^n$ in $\overline{\mathbb{C}}$, such that $\bigcup B_n$ has no limit points in $D$. Let $\alpha_n$ and $\beta_n$ be the normalized point counting measures of $A_n$ and $B_n$ respectively and $\alpha_n'$ and $\beta_n'$ their swept measures onto $\partial D$. Denote by $U_\mu$ the logarithmic potential of the measure $\mu$. It has been shown by Ambroladze and Wallin, that if for every weak-star limit point $\alpha$ of $\{\alpha_n\}$, we have $\alpha (D)>0$ and also $$ \lim_{n\to\infty}\left[\sup_{z\in\partial D} \left(\frac{n+1}{n}U_{\alpha_n'}(z)-U_{\beta_n'}(z)\right) \right]=0, $$ then for every bounded analytic function $f$ on $D$, the rational interpolants $r_n$ of degree $n$ with poles at $B_n$ interpolating to $f$ at $A_n$, converge locally uniformly with geometric degree of convergence to $f$ on $D$. We show that under a slightly stricter condition on $\{B_n\}$, the boundedness requirement on~$f$ can be replaced with the weaker growth condition $$ |f(z)|\leq Ad(z,\partial D)^q, $$ where $A>0$, $q<0$ and that this growth condition is also sharp in the sense described in this paper.
Keywords: Rational interpolation, logarithmic potentials.
MSC 2000: Primary 30E10; Secondary 31A15.
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