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 $\mathbb {C}$ 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]

References

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

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

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

[1]

Theorem

See also

References