Extension of Arakelyan's Theorem

Spyros Pasias

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arXiv · 2304.03334

Extension of Arakelyan's Theorem

Spyros Pasias

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Abstract

Arakeljan's Theorem provides conditions on a relatively closed subset $F$ of a domain $G\subset\mathbb{C}$, such that any continuous function $f:F\rightarrow\mathbb{C}$ that is analytic in $F^\circ$, can be approximated by analytic functions defined on $G$. In this paper we will extend Arakeljan's theorem by adding the extra requirement that the analytic functions that approximate $f$ may also be chosen to be bounded on a closed set $C\subset G.$ In \cite{RU} the same problem has been considered but for the specific case that $G=\mathbb{C}$. In this paper we will extend the result in \cite{RU} and show that is true for an arbitrary $G$, provided that $F$ and $C$ satisfy certain topological condition in $G$. Additionally, we will show that the result holds always true when $G$ is simply connected.

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Spyros Pasias. 2023-04-06. Extension of Arakelyan's Theorem. https://arxiv.org/abs/2304.03334

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