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arXiv · 2512.00802

A logarithmic characterization of Arakelian sets

Grigorios Fournodavlos·Vassili Nestoridis·Spyros Pasias
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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.

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Grigorios Fournodavlos, Vassili Nestoridis, Spyros Pasias. 2025-11-30. A logarithmic characterization of Arakelian sets. https://doi.org/10.1134/s1995080224604429

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