An approximation theorem related to good compact sets in the sense of Martineau. (English) Zbl 0964.32010

Rosay, Jean-Pierre; Stout, Edgar Lee

Ann. Inst. Fourier 50, No. 2, 677-687 (2000); corrigendum ibid. 52, No. 4, 1285 (2002).

The goal of the paper is to give a complete proof of the following statement.
Theorem. Let \(K\) be a compact set in an open set \(V\) of \(\mathbb{C}^n\). Then there exists a neighborhood \(W\) of \(K\) such that every holomorphic function on \(U\), which is uniformly approximable by polynomials on some neighborhood of \(K\), is uniformly approximable by polynomials on \(W\).
The authors claim that this result published by J. E. Björk in 1970 [Ann. Inst. Fourier 20, 493-498 (1970; Zbl 0188.39003)] is not fully proved by him. They give a proof using a result of E. Bishop, which they also prove for the convenience of the reader.

Reviewer: Sergey M.Ivashkovich (Villeneuve d’Ascq)

MSC:

32E30	Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
32E20	Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
46A22	Theorems of Hahn-Banach type; extension and lifting of functionals and operators

Keywords:

polynomial hulls; polynomial approximation; analytic functionals; good compact sets

Citations:

Zbl 0188.39003

References:

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[2]	J.E. BJÖRK, Every compact set in ℂn is a good compact set, Ann. Inst. Fourier, Grenoble, 20-1 (1970), 493-498.0188.39003AIF_1970__20_1_493_0 · Zbl 0188.39003
[3]	R.C. GUNNING, Introduction to Holomorphic Functions of Several Complex Variables, vol. I, Wadsworth and Brooks-Cole, Belmont, 1990. · Zbl 0699.32001
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