Universal Taylor series on doubly connected domains with respect to every center
In this paper, the author studies overconvergence phenomena with respect to different centers in the domain of holomorphy, which had been primarily investigated by \textit{W.~Luh} [Analysis 6, 191--207 (1986; Zbl 0589.30003)]. Let \(\Omega\) be an open subset of the complex plane \(\mathbb C\) and \(H(\Omega )\) be the Fréchet space of all holomorphic functions on \(\Omega\). If \(f \in H(\Omega )\) and \(t \in \Omega\), let \(S_n(f,t)\) be the \(n\)th partial sum of the Taylor development of \(f\) with center \(t\). By definition, a function \(f \in H(\Omega )\) is said to be a universal Taylor series at \(t\) in the sense of Nestoridis -- its set is denoted by \(U(\Omega ,t)\) -- if for every compact subset \(K \subset \mathbb C \setminus \Omega\) with \(K^c\) connected and for every function \(h:K \to \mathbb C\), continuous on \(K\) and holomorphic in \(K^0\), there exists a sequence \((n_j)\) of natural numbers such that \(\lim_{n \to \infty} \sup_{z \in K} | S_{n_j}(f,t)(z) - h(z)| = 0\). A function \(f \in H(\Omega )\) is said to belong to the class \(U(\Omega )\) if, for every pair \(K,h\) as before, there is a sequence \((n_j)\) of natural numbers such that for every \(L \subset \Omega\) compact the following holds: \(\lim_{n \to \infty} \sup_{t \in L} \sup_{z \in K} | S_{n_j}(f,t)(z) - h(z)| = 0\). In the early 70s Luh and, independently, Chui and Parnes, gave a similar definition to \(U(\Omega ,t)\), where the compact set is not allowed to contain pieces of the boundary of \(\Omega\); this class is denoted by \(U_1(\Omega ,t)\). By \(U_1(\Omega )\) it is denoted the corresponding analogous class to \(U(\Omega )\). Hence \(U(\Omega ) \subset U_1(\Omega )\), \(U(\Omega ,t) \subset U_1(\Omega ,t)\), \(U(\Omega ) \subset U(\Omega ,t)\) and \(U_1(\Omega ) \subset U_1(\Omega ,t)\). It holds that \(U(\Omega ) = \emptyset\) if \(\Omega\) is a non-simply connected domain, and that \(U_1(\mathbb C \setminus \overline{D}) = \emptyset\), where \(D\) is the open unit disk. The author proves that, in spite of the last facts, we have: (a) If \(\Omega = C \setminus K\), where \(K\) is a closed polygonal line with its interior, then the class \(\bigcap_{t \in \Omega} U(\Omega ,t)\) is residual in \(H(\Omega )\), hence non-empty. (b) The class \(\bigcap_{t \in C \setminus \overline{D}} U_1(C \setminus \overline{D},t)\) is residual in \(H(\Omega )\), hence non-empty.
scientific article; zbMATH DE number 2174966