Universal rational expansions of meromorphic functions
Let \(G \subsetneq \mathbb{C}\) be a domain. The author proves that every meromorphic function \(\phi\) on \(G\) possesses a universal rational expansion, i.e., there exists a series \(\sum_{\nu=1}^\infty R_\nu(z)\) of rational functions, such that the partial sums \(\phi_n(z):= \sum_{\nu=1}^n R_\nu(z)\) have the following properties:{\parindent=8mm\begin{itemize}\item[(i)] \(\phi_n\) converges to \(\phi\) locally uniformly on \(G\) with respect to the chordal metric.\item[(ii)] For each compact set \(K \subset G^c\), each function \(f\) holomorphic on \(K\), and each \(l \in \mathbb{N}_0\), there is a suitable subsequence \(\phi_{n_k}^{(l)}(z)\) which converges uniformly on \(K\) to \(f^{(l)}(z)\), the \(l\)-th derivative of \(f\).\item[(iii)] For each compact set \(K \subset G^c\) and each function \(g\) meromorphic on \(K\), there is a suitable subsequence \(\phi_{m_k}(z)\) which converges uniformly on \(K\) to \(g(z)\) with respect to the chordal metric. \end{itemize}} The paper concludes with the nice result, that the derivative of a universal rational expansion \(\sum_{\nu=1}^\infty R_\nu(z)\) is not universal in general. If \(\mathbb{C} \setminus G\) has non-empty interior, then \(\sum_{\nu=1}^\infty R_\nu'(z)\) is not a universal rational expansion of any meromorphic function.
scientific article; zbMATH DE number 5965044