A logarithmic characterization of Arakelian sets

Abstract

Arakelian's classical approximation theorem \cite{Ar} gives necessary and sufficient conditions such that functions can be uniformly approximated in (unbounded) closed sets $F\subset \mathbb{C}$ by entire functions. The conditions are purely topological and concern the connectedness of the complement of $F$. We give a new characterization of Arakelian sets in terms of logarithmic branches of functions $f\in A(F)$, which are continuous in $F$ and holomorphic in its interior $F^{\circ }$. Our proof is based on a contradiction argument and the counterexample function that we use is furnished by the Weierstrass factorization theorem.