A generic result concerning univalent universal functions
Assume that \(G\) is a Jordan domain in the complex plane \(\mathbb C\), and that \(K \subset \mathbb C\) is a compact set. By \(H(G)\), \(A(G)\), \(A(K)\), \(B(G)\) it is denoted, respectively, the space of all holomorphic functions on \(G\), the space of continuous functions on \(\overline{G}\) that are holomorphic in \(G\), the space of continuous functions on \(K\) that are holomorphic in \(K^0\), and the subspace of \(A(G)\) consisting of functions that are either univalent or constant on \(G\). If \(A(G)\) is endowed with the maximum norm, then it becomes a Banach space and \(B(G)\) is closed in it, so \(B(G)\) is a Baire space. In addition, if \(a \in G\) then \(W\) denotes the class of holomorphic functions that are universal under derivatives and universal with respect to overconvergence, and performing both approximations with the same approximative sequence, that is, \(f \in W\) whenever \(f \in H(G)\) and, for every choice of compact sets \(L \subset G\) and \(K \subset \overline{G}^c\) with \(L^c\) and \(K^c\) connected and for every choice of functions \(\phi \in A(K)\), \(h \in A(L)\), there exists a sequence \((n_j)\) of natural numbers such that \(\lim_{j \to \infty} \sup_{z \in K}| S_{n_j}(f,a)(z) - \phi (z)| = 0 = \lim_{j \to \infty} \sup_{z \in L} | f^{(n_j)}(z) - h(z)| \), where \(S_n(f,a)\) is the \(n\)th-partial sum of the Taylor expansion of \(f\) with center \(a\). By using a modified version of a classical result of J.~Walsh concerning approximation of holomorphic functions by polynomials, the authors prove that \(W \cap B(G)\) is residual in \(B(G)\). Hence there exist functions in \(A(G)\) that are universal (in both senses of derivation and overconvergence) and also univalent in \(G\).
universal with respect to overconvergence
scientific article; zbMATH DE number 2106409