Sets in ${\mathbb{C}}^N$ with vanishing global extremal function and polynomial approximation

Józef Siciak

doi:10.5802/afst.1312

Sets in

ℂ^{N}

with vanishing global extremal function and polynomial approximation

Józef Siciak ¹

¹ Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. S2, pp. 189-209.

Abstract
Abstract

Let $Γ$ be a non-pluripolar set in $ℂ^{N}$ . Let $f$ be a function holomorphic in a connected open neighborhood $G$ of $Γ$ . Let ${P_{n}}$ be a sequence of polynomials with $deg P_{n} \leq d_{n} (d_{n} < d_{n + 1})$ such that

\underset{n \to \infty}{lim sup} {| f (z) - P_{n} (z) |}^{1 / d_{n}} < 1, z \in Γ .

We show that if

\underset{n \to \infty}{lim sup} {| P_{n} (z) |}^{1 / d_{n}} \leq 1, z \in E,

where $E$ is a set in $ℂ^{N}$ such that the global extremal function $V_{E} \equiv 0$ in $ℂ^{N}$ , then the maximal domain of existence $G_{f}$ of $f$ is one-sheeted, and

\underset{n \to \infty}{lim sup} {∥ f - P_{n} ∥}_{K}^{\frac{1}{d_{n}}} < 1

for every compact set $K \subset G_{f}$ . If, moreover, the sequence ${d_{n + 1} / d_{n}}$ is bounded then $G_{f} = ℂ^{N}$ .

If $E$ is a closed set in $ℂ^{N}$ then $V_{E} \equiv 0$ if and only if each series of homogeneous polynomials $\sum_{j = 0}^{\infty} Q_{j}$ , for which some subsequence ${s_{n_{k}}}$ of partial sums converges point-wise on $E$ , possesses Ostrowski gaps relative to a subsequence ${n_{k_{l}}}$ of ${n_{k}}$ .

In one-dimensional setting these results are due to J. Müller and A. Yavrian [5].

Soit $Γ$ un sous-ensemble non pluripolaire de $ℂ^{N}$ . Soit $f$ une fonction holomorphe sur un voisinage ouvert connexe $G$ de $Γ$ . Soit ${P_{n}}$ une suite de polynômes de degré $deg P_{n} \leq d_{n} (d_{n} < d_{n + 1})$ telle que

\underset{n \to \infty}{lim sup} {| f (z) - P_{n} (z) |}^{1 / d_{n}} < 1, z \in Γ .

On démontre que si

\underset{n \to \infty}{lim sup} {| P_{n} (z) |}^{1 / d_{n}} \leq 1, z \in E,

où $E$ is est un sous-ensemble de $ℂ^{N}$ tel que la fonction extrémale globale $V_{E} \equiv 0$ sur $ℂ^{N}$ , alors le domaine maximal d’existence $G_{f}$ de $f$ est uniforme, et

\underset{n \to \infty}{lim sup} {∥ f - P_{n} ∥}_{K}^{\frac{1}{d_{n}}} < 1

pour tout compact $K \subset G_{f}$ . Si, de plus, la suite ${d_{n + 1} / d_{n}}$ est bornée alors $G_{f} = ℂ^{N}$ .

Si $E$ est un sous-ensemble fermé de $ℂ^{N}$ alors $V_{E} \equiv 0$ si et seulement si chaque série de polynômes homogènes $\sum_{j = 0}^{\infty} Q_{j}$ , ayant une sous-suite ${s_{n_{k}}}$ de sommes partielles convergeant ponctuellement sur $E$ , admet des lacunes de type Ostrowski relativement à une sous-suite ${n_{k_{l}}}$ de ${n_{k}}$ .

En dimension $1$ , ces résultats sont dûs à J. Müller and A. Yavrian [5].

MR Zbl

DOI: 10.5802/afst.1312

Author's affiliations:

Józef Siciak ¹

¹ Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

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     author = {J\'ozef Siciak},
     title = {Sets in ${\mathbb{C}}^N$ with vanishing global extremal function and polynomial approximation},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {189--209},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 20},
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     year = {2011},
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Józef Siciak. Sets in ${\mathbb{C}}^N$ with vanishing global extremal function and polynomial approximation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. S2, pp. 189-209. doi : 10.5802/afst.1312. https://afst.centre-mersenne.org/articles/10.5802/afst.1312/

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