References & Citations
Mathematics > Complex Variables
Title: Extension of Arakelyan's Theorem
(Submitted on 6 Apr 2023 (v1), last revised 29 Apr 2024 (this version, v6))
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.
Submission history
From: Spyros Pasias Ph.D [view email][v1] Thu, 6 Apr 2023 19:05:03 GMT (9kb)
[v2] Tue, 27 Jun 2023 15:19:36 GMT (9kb)
[v3] Fri, 3 Nov 2023 18:19:59 GMT (9kb)
[v4] Sat, 18 Nov 2023 14:16:24 GMT (9kb)
[v5] Fri, 26 Apr 2024 10:34:40 GMT (9kb)
[v6] Mon, 29 Apr 2024 17:08:05 GMT (9kb)
Link back to: arXiv, form interface, contact.