Generic approach to multiply universal functions

Kurzfassung/Abstract

Let $\{ z_n \}$ be a sequence of complex numbers. The author uses a generic approach to show the existence of universal entire functions $\varphi$ having several universal properties, i.e. for any preassigned compact set $K \subset \mathbb{C}$ with connected complement and any preassigned function $f \in C(K)$, holomorphic in the interior of $K$, there exist \par 1) a suitable sequence $\varphi(z + z_{n_k^{'}})$ of additively translates,\par 2) a suitable sequence $\varphi(z_{n_k^{}} \cdot z)$ of multiplicatively translates (if $0 \notin K$), and \par 3) a suitable sequence $\varphi^{([|z_{n_k^{'}}|])}(z)$ of derivatives \par of $\varphi$ that all converge to $f$ uniformly on $K$. By a so-called constructive approach, {\it W. Luh} [Complex Variables, Theory Appl. 31, 87--96 (1996; Zbl 0869.30022)] obtained functions of this type. In addition, the constructed functions $\varphi$ join two non-universal properties, namely having zeros at certain prescribed points of prescribed order and solving a certain interpolation problem.