Universal Taylor series on doubly connected domains
Let \(K\subset\mathbb{C}\) be a connected compact set such that \(\Omega=\mathbb{C}\setminus K\) is connected, and \(\zeta_0\in\Omega\). Also, let \(U^\infty(\Omega,\zeta_0)\) denote the class of holomorphic functions \(f:\Omega\to\mathbb{C}\) said to be `universal' with respect to \(\Omega\) and \(\zeta_0\) if the following holds: for every polynomial \(h\), there exists a sequence \(\{\lambda_n\}\) of non-negative integers such that, for all \[ l=0,1,2,\dots,\;\lim_{n\to\infty}\left(\sup_{z\in K}\left| \frac{\partial^l}{\partial z^l}S_{\lambda_n}(f,\zeta_0)(z)-h^{(l)}(z) \right|\right)=0. \] The existence of universal Taylor series was established by \textit{V. Nestoridis} [Ann. Inst. Fourier 46, No. 5, 1293--1306 (1996; Zbl 0865.30001)]. The author proves that, for every \(\zeta_0\in\Omega\), the class \(U^\infty(\Omega,\zeta_0)\) is \(G_\delta\) and dense in \(H(\Omega)\) and so non-empty; and \(U^\infty(\Omega, \zeta_0)\{0\}\) contains a dense vector subspace of \(H(\Omega)\). (This extends an earlier result of \textit{A. Melas} [Ann. Inst. Fourier 51, 1539--1551 (2001; Zbl 0989.30003)]). He also proves the same results in the subspace \(H_0(\Omega)\) of \(H(\Omega)\) which contains those functions \(f\in H(\Omega)\) for which \(\lim_{z\to\infty}(z)=0\).
scientific article; zbMATH DE number 5565946