A logarithmic characterization of Arakelian sets

TL;DR

This paper introduces a new logarithmic-based characterization of Arakelian sets, linking topological conditions with the behavior of logarithmic branches of functions, enhancing understanding of approximation by entire functions.

Contribution

It provides a novel characterization of Arakelian sets using logarithmic branches, expanding the theoretical framework of complex approximation theory.

Findings

New characterization of Arakelian sets via logarithmic branches

Connection between topological connectedness and logarithmic function behavior

Use of Weierstrass factorization theorem in proof

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.