A logarithmic characterization of Arakelian sets
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917114497990656 |
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| author | Fournodavlos, Grigorios Nestoridis, Vassili Pasias, Spyros |
| author_facet | Fournodavlos, Grigorios Nestoridis, Vassili Pasias, Spyros |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00802 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A logarithmic characterization of Arakelian sets Fournodavlos, Grigorios Nestoridis, Vassili Pasias, Spyros Complex Variables 30E10 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. |
| title | A logarithmic characterization of Arakelian sets |
| topic | Complex Variables 30E10 |
| url | https://arxiv.org/abs/2512.00802 |