Boundedness, regularity and smoothness of universal Taylor series
The author improves results of \textit{G.~Costakis} [Math. Proc. Camb. Philos. Soc. 128, 157--175 (2000; Zbl 0956.30003)] on the existence of universal holomorphic functions. To be more precise, let \(\Omega \subset \mathbb C\) be an unbounded simply connected domain such that \((\overline{\Omega})^\circ = \Omega\), \((\mathbb C \setminus \overline{\Omega}) \cup \{\infty\}\) is connected and locally connected at \(\infty\). For example, \(\Omega\) might be an open half plane. Furthermore, let \(A(\Omega)\) denote the space of all functions which are continuous on \(\overline{\Omega}\) and holomorphic in \(\Omega\), and let \(H^\infty(\Omega)\) denote the space of all functions which are holomorphic and bounded in \(\Omega\). Finally, let \(X = A(\Omega) \cap H^\infty(\Omega)\). Then \(X\) equipped with the supremum norm on \(\overline{\Omega}\) is a Banach space. For \(f \in X\) and \(\zeta \in \Omega\), let \[ S_n(f,\zeta)(z) = \sum_{k=0}^n \frac{f^{(k)}(\zeta)}{k!}\,(z-\zeta)^k \] be the \(n\)-th partial sum of the Taylor series of \(f\) around the point \(\zeta\). A function \(f \in X\) is called universal in the weak sense if for every compact set \(K \subset \mathbb C\) with \(K \cap \overline{\Omega} = \varnothing\), \(K^c\) connected and for every function \(h\) continuous on \(K\) and holomorphic in \(K^\circ\), there exists a sequence \((\lambda_n)\) of nonnegative integers such that for every compact set \(L \subset \Omega\) there holds \[ \lim_{n\to\infty}{\sup_{\zeta \in L}{\sup_{z \in K}{| S_{\lambda_n}(f,\zeta)(z)-h(z)| }}} = 0\,. \] The author proves that the set \(U_1\) of all these universal functions is a dense \(G_\delta\) set in \(X\). In particular \(U_1 \neq \varnothing\). Also the author proves similar results on universal functions vanishing at \(\infty\) and which are smooth on the boundary of \(\Omega\) as well as results on universal functions in the strong sense (which means that \(K \cap \Omega = \varnothing\)). The main tools of the proofs are Baire's theorem and Arakeljan's approximation theorem.
scientific article; zbMATH DE number 5077606
scientific article