Universal Meromorphic Approximation on Vitushkin Sets

Kurzfassung/Abstract

The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence $\{\lambda_n\in\cup\mathb{C}$, there exists a function $\Phi$, meromorphic on $\mathbb{C}$, with the following property. For every compact set $K$ of rational approximation (i.e. Vitushkin set), and every function $f$, continuous on $K$ and holomorphic in the interior of $K$, there exists a subsequence $\{n_k\}$ of $\mathbb{N}$ such that $ \left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\}$ converges to $f(z)$ uniformly on $K$.
A similar result is obtained for arbitrary domains $G \neq \mathbb{C}$. Moreover, in case $\{\lambda_n\}=\{n\}$ the function $\Phi$ is frequently universal in terms of Bayart/Grivaux [3].