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

##### 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
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##### References:
 [1] E. BISHOP, Holomorphic completions, analytic continuations, and the interpolation of semi-norms, Ann. Math., 78 (1963), 468-500. · Zbl 0131.30901 [2] J.E. BJÖRK, Every compact set in ℂn is a good compact set, Ann. Inst. Fourier, Grenoble, 20-1 (1970), 493-498. · 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 [4] R.C. GUNNING and H. ROSSI, Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, 1965. · Zbl 0141.08601 [5] A. MARTINEAU, Sur LES fonctionnelles analytiques et la transformation de Fourier-Borel, J. Analyse Math., XI (1963), 1-164. (Also contained in the Œuvres of Martineau). · Zbl 0124.31804 [6] J.-P. ROSAY and E.L. STOUT, Strong boundary values, analytic functionals and nonlinear Paley-Wiener theory, to appear. · Zbl 0988.46032 [7] W.R. ZAME, Algebras of analytic germs, Trans. Amer. Math. Soc., 174 (1972), 275-288. · Zbl 0267.32009
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