Arakelyan's theorem
{{Short description|Mathematical theorem}}
In mathematics, Arakelyan's theorem is a generalization of Mergelyan's theorem from compact subsets of an open subset of the complex plane to relatively closed subsets of an open subset.
Theorem
Let Ω be an open subset of and E a relatively closed subset of Ω. By Ω* is denoted the Alexandroff compactification of Ω.
Arakelyan's theorem states that for every f continuous in E and holomorphic in the interior of E and for every ε > 0 there exists g holomorphic in Ω such that |g − f| < ε on E if and only if Ω* \ E is connected and locally connected.{{cite book|last1=Gardiner|first1=Stephen J.|title=Harmonic approximation|url=https://archive.org/details/harmonicapproxim00gard_738|url-access=limited|date=1995|publisher=Cambridge University Press|location=Cambridge|isbn=9780521497992|page=[https://archive.org/details/harmonicapproxim00gard_738/page/n50 39]}}
See also
References
{{reflist}}
- {{cite journal|last1=Arakeljan|first1=N. U.|authorlink=Norair Arakelian|title=Uniform and tangential approximations by analytic functions|journal=Izv. Akad. Nauk Armjan. SSR Ser. Mat|date=1968|volume=3|pages=273–286}}
- {{cite book|last1=Arakeljan|first1=N. U|title=Actes, Congrès intern. Math.|date=1970|volume=2|pages=595–600}}
- {{cite journal|last1=Rosay|first1=Jean-Pierre|last2=Rudin|first2=Walter|title=Arakelian's Approximation Theorem|journal=The American Mathematical Monthly|date=May 1989|volume=96|issue=5|pages=432|doi=10.2307/2325151|jstor=2325151}}