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.

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 |

##### 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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