On uniform convergence of rational, Newton-Padé interpolants of type (n,n) with free poles as \(n\to \infty\) (Q1826048)

Let f be a meromorphic function in the complex plane \({\mathbb{C}}\). The author shows that there exists a sequence of distinct interpolation points \(\{z_ i\}_{i=1,2,...}\), and for \(n\geq 1\), rational functions \(R_ n(z)\) of type (n,n) solving the Newton-Padé (Hermite) interpolation problem, \(R_ n(z_ i)=f(z_ i),\) \(i=1,2,...,2n+1\), such that for each compact subset K of \({\mathbb{C}}\) except the poles of f, it holds that \(\lim_{n\to \infty}\| f-R_ n\|^{1/n}_{L_{\infty}(K)}=0.\) And the author discusses some extended and related results in the case that f(z) is meromorphic in a given open set with certain additional conditions.