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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

MSC:
 3.2e+31 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
Runge exhaustion
Full Text:
References:
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