Arakelyan's 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 ℂ 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

References

  1. Gardiner, Stephen J. (1995). Harmonic approximation. Cambridge: Cambridge University Press. p. 39. ISBN 9780521497992.
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