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Runge approximation. (English) Zbl 0636.32006
The paper deals with the problem of Runge exhaustion of smoothly bounded domains. Especially the following local question is investigated.
If D is an open domain in \({\mathbb{C}}^ n,\) \(D=\{q\in {\mathbb{C}}^ n:\) \(r(q)<0\}\), where r is a smooth defining function and \(p\in \partial D\), does there exist a neighbourhood U of P and an \(\epsilon_ 0>0\) so that the following holds: The domain \(D_ s=\{q\in {\mathbb{C}}^ n:\) \(s(q)<0\}\) is Runge in D whenever \(s\geq r\), \(s=r\) outside U and \(\| s- r\|_{C^ 2}<\epsilon\) where \(\epsilon <\epsilon_ 0\), (s is a \(C^{\infty}\)-function)?
Recall that the envelope of holomorphy \(\tilde D\) of D is schlicht if it can be realized as an open domain in \({\mathbb{C}}^ n.\)
Theorem 1.4. Let D be a smoothly bounded domain in \({\mathbb{C}}^ n,\) \(D=\{q\in {\mathbb{C}}^ n:\) \(r(q)<0\}\). Assume that \(\tilde D\) is schlicht. Then \(D_ s\) is Runge in D if \(\epsilon_ 0>0\) and U is small enough and if \(p\in \partial D\) is of type 1 or 2. If p is of type 3, then \(D_ s\) is not always Runge in D.
The point \(p\in \partial D\) is said to be of type 1 if there exists a neighbourhood W of p so that \(W\cap \partial D\subset W\cap \partial \tilde D\); p is of type 2 if \(p\in \tilde D\); p is of type 3 if for any neighbourhood V of p V contains points from \(\partial \tilde D\) and from \(\tilde D\).
Reviewer: S.M.Ivashkovich

32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
Full Text: DOI EuDML
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