Rational and polynomial interpolation of analytic functions with restricted growth (Q5958887)

Consider a domain \(D\subset \overline \mathbb{C}=\mathbb{C}\cup \{\infty\}\) and an analytic function \(f\) on \(D\). Let for each \(n\geq 1\), \(A_n= \{a_{ni}\}^n_{i=0} \subset D\) and \(B_n=\{b_{n_i}\}^n_{i=1} \subset\mathbb{C}\) be two collections of points (some \(a_{n_i}\) or some \(b_{n_i}\) may coincide). Then there exists a unique rational function \(r_n\), of order \(n\) (the degree of numerator and the degree of denominator are at most \(n)\), with poles at \(B_n\), interpolating to \(f\) at \(A_n\), counting multiplicities. If \(A_n\) and \(B_n\) have some common points, we cancel them in the construction of \(r_n\). In this paper, the author examines the question: Given \(D\), is it possible to choose poles \(B_n\) and interpolating points \(A_n\) in such a way that for all analytic \(f\), \(r_n\) converges to \(f\). Uniformly on compact subsets of \(D\), as \(n\to \infty\)?