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Analytic continuation from compact set. (English) Zbl 0664.32009

Let K be a compact set in \(K^ n={\mathbb{R}}^ n\) or \({\mathbb{C}}^ n\) and let A(K) be a space of germs on K of all functions analytic near K. For a point \(x\in K\) and germ \(f_ K\) in A(K) let \(T_ xf_ K\) denote the power series expansion at the point x of germ \(f_ K\) whose radius of convergence is \(r(T_ xf_ K)\). Let further, for \(R>0\) \[ A_ R(K)=\{f_ K\in A(K):\quad r(T_ xf_ K)\geq R,\quad \forall x\in K\}. \] We say that a compact K has the extension property if for every \(R>0\) exist an open neighbourhood \(U_ k\) of K such that \(A_ R(K)\subset A(U_ R).\)
Using the geometric approach the author proofs the following theorems:
Theorem 3.1. Every continuum \(K\subset R^ 2\) has the extension property.
Theorem 4.2. Every L-connected (in sense of Zame) continuum \(K\subset K^ n\) has the extension property.
These results permits to the author to obtain in a unified manner the results of G. T. Varfolomeev, W. R. Zame and J. T. Rogers.
Reviewer: N.I.Skiba

MSC:

32D15 Continuation of analytic objects in several complex variables
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