Arakelyan's theorem explained

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

\Complex 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.^[1]

See also

• Runge's theorem
• Mergelyan's theorem

References

• Arakeljan. N. U.. Norair Arakelian. Uniform and tangential approximations by analytic functions. Izv. Akad. Nauk Armjan. SSR Ser. Mat. 1968. 3. 273–286.
• Book: Arakeljan. N. U. Actes, Congrès intern. Math.. 1970. 2. 595–600.
• Rosay. Jean-Pierre. Rudin. Walter. Arakelian's Approximation Theorem. The American Mathematical Monthly. May 1989. 96. 5. 432. 10.2307/2325151. 2325151.

Notes and References

1. Book: Gardiner. Stephen J.. Harmonic approximation. limited. 1995. Cambridge University Press. Cambridge. 9780521497992. 39.