Universal Taylor series have a strong form of universality
The paper deals with the class \(U(\Omega,\xi)\) of holomorphic in a simply connected domain \(\Omega\) functions \(f\) whose Taylor series at \(\xi \in \Omega\) is universal with respect to overconvergence. It is considered the expansion of \(f\) in Faber series corresponding to any connected compact set \(\Gamma \subset \Omega\) with more than one point and such that \(\widehat{\mathbb{C}}\setminus \Gamma\) is connected (here \(\widehat{\mathbb{C}}\) denotes the Riemann sphere): \[ f(z)=\sum_{n=0}^{+\infty} c_n(f,\Gamma)F_n(\Gamma,z). \] Let \(S_n(f,\Gamma)(z)=\sum_{\nu=0}^{n} c_{\nu}(f,\Gamma)F_{\nu }(\Gamma,z)\) be a partial sum of the series. Definition 1. A holomorphic function \(f\in H(\Omega)\) belongs to the class \(U(\Omega,\Gamma)\) provided that for every compact set \(K\subset\mathbb{C}\) with \(K\cap\Omega=\emptyset\) and \(K^{\mathbb{C}}\) connected and every function \(h:K \to \mathbb{C}\) continuous on \(K\) and holomorphic in \(K^0\), there exists a sequence \((\lambda _n)\) of natural numbers such that \(S_{\lambda_n}(f,\Gamma)(z)\) converges to \(h(z)\) uniformly on \(K\) as \(n \to +\infty .\) Definition 2. A holomorphic function \(f\in H(\Omega)\) belongs to the class \(U_{\text{Fab}}(\Omega)\) provided that for every compact set \(K\subset\mathbb{C}\) with \(K\cap\Omega=\emptyset\) and \(K^{\mathbb{C}}\) connected and every function \(h:K \to \mathbb{C}\) continuous on \(K\) and holomorphic in \(K^0\), there exists a sequence \((\lambda _n)\) of natural numbers such that, for any connected compact set \(M\subset \Omega\) with more than one point and such that \(\widehat{\mathbb{C}}\setminus M\) is connected \[ \sup_{\Gamma \in M }\sup_{z \in K}| S_{\lambda_n}(f,\Gamma)(z)-h(z)| \to 0, \quad n \to +\infty. \] The authors obtain the following result, which answers a question of Mayenberger and Müller. Theorem. \(U(\Omega,\xi)=U(\Omega,\Gamma)=U_{\text{Fab}}(\Omega).\) The topology used here is the Carathéodory topology.
scientific article; zbMATH DE number 5320470