On the rate of rational approximation to functions with a finite number of poles
Let K be a compact subset of \({\mathbb{C}}\) bounded by a Jordan curve C. Let \(\Gamma_ R=\phi (\{| z| =R\})\) \((R>1)\) be the level line of the corresponding conformal mapping \(\phi\) of \(\{| z| >1\}\) onto \({\hat {\mathbb{C}}}\setminus K\), and let \(E_ R\) be the interior of \(\Gamma_ R\), such that \(K\subset E_ R\). Let f be meromorphic on \(E_ R\) with precisely \(\nu\) poles. For the case that C is sufficiently smooth (more exactly: if K is a Faber set), the author describes the asymptotic behaviour of \[ \max_{w\in K}| f(w)-R_{n\nu}(w)|, \] where \(R_{n\nu}\) is the best rational approximation (admitting \(\nu\) poles) to f on K, in terms of the growth of f. Further, he extends his results to the case of interpolation and investigates some other related topics.
scientific article; zbMATH DE number 4181719
scientific article