2023/04/06 by Spyros Pasias, Pasias, Spyros
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2304.03334
openalex publication_date 2023/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Arakeljan's Theorem provides conditions on a relatively closed subset F of a domain G⊂ℂ, such that any continuous function f:F→ℂ that is analytic in F^∘, 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⊂ G. In \citeRU the same problem has been considered but for the specific case that G=ℂ. In this paper we will extend the result in \citeRU 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.