Extension of Arakelyan's Theorem
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913339731345408 |
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| author | Pasias, Spyros |
| author_facet | Pasias, Spyros |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_03334 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extension of Arakelyan's Theorem Pasias, Spyros Complex Variables 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. |
| title | Extension of Arakelyan's Theorem |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2304.03334 |