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

Siciak, Józef. "Sets in ${\mathbb{C}}^N$ with vanishing global extremal function and polynomial approximation." Annales de la faculté des sciences de Toulouse Mathématiques 20.S2 (2011): 189-209. <http://eudml.org/doc/219851>.

@article{Siciak2011,
abstract = {Let $\Gamma $ be a non-pluripolar set in $\{\mathbb\{C\}\}^N$. Let $f$ be a function holomorphic in a connected open neighborhood $G$ of $\Gamma $. Let $\lbrace P_n\rbrace $ be a sequence of polynomials with $\deg P_n\,\le \, d_n \,\,(d_n&lt;d_\{n+1\})$ such that\[\limsup \_\{n\rightarrow \infty \}|f(z)-P\_n(z)|^\{1/d\_n\}\,\,&lt;\,1,\,\,z\in \Gamma .\]We show that if\[\limsup \_\{n\rightarrow \infty \}|P\_n(z)|^\{1/d\_n\} \,\le \,1,\,\,z\in E,\]where $E$ is a set in $\{\mathbb\{C\}\}^N$ such that the global extremal function $V_E\,\equiv \,0$ in $\{\mathbb\{C\}\}^N$, then the maximal domain of existence $G_f$ of $f$ is one-sheeted, and\[\limsup \_\{n\rightarrow \infty \}\Vert f-P\_n\Vert \_K^\{\frac\{1\}\{d\_n\}\}\,&lt;\,1\]for every compact set $K\subset G_f$. If, moreover, the sequence $\lbrace d_\{n+1\}/d_n\rbrace $ is bounded then $G_f\,=\,\{\mathbb\{C\}\}^N$.If $E$ is a closed set in $\{\mathbb\{C\}\}^N$ then $V_E\,\equiv \, 0$ if and only if each series of homogeneous polynomials $\sum _\{j=0\}^\{\infty \}Q_j$, for which some subsequence $\lbrace s_\{n_k\}\rbrace $ of partial sums converges point-wise on $E$, possesses Ostrowski gaps relative to a subsequence $\lbrace n_\{k_l\}\rbrace $ of $\lbrace n_k\rbrace $.In one-dimensional setting these results are due to J. Müller and A. Yavrian [5].},
affiliation = {Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland},
author = {Siciak, Józef},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Taylor seies; Ostrowski gaps; Siciak extremal function},
language = {eng},
month = {4},
number = {S2},
pages = {189-209},
publisher = {Université Paul Sabatier, Toulouse},
title = {Sets in $\{\mathbb\{C\}\}^N$ with vanishing global extremal function and polynomial approximation},
url = {http://eudml.org/doc/219851},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Siciak, Józef
TI - Sets in ${\mathbb{C}}^N$ with vanishing global extremal function and polynomial approximation
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/4//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - S2
SP - 189
EP - 209
AB - Let $\Gamma $ be a non-pluripolar set in ${\mathbb{C}}^N$. Let $f$ be a function holomorphic in a connected open neighborhood $G$ of $\Gamma $. Let $\lbrace P_n\rbrace $ be a sequence of polynomials with $\deg P_n\,\le \, d_n \,\,(d_n&lt;d_{n+1})$ such that\[\limsup _{n\rightarrow \infty }|f(z)-P_n(z)|^{1/d_n}\,\,&lt;\,1,\,\,z\in \Gamma .\]We show that if\[\limsup _{n\rightarrow \infty }|P_n(z)|^{1/d_n} \,\le \,1,\,\,z\in E,\]where $E$ is a set in ${\mathbb{C}}^N$ such that the global extremal function $V_E\,\equiv \,0$ in ${\mathbb{C}}^N$, then the maximal domain of existence $G_f$ of $f$ is one-sheeted, and\[\limsup _{n\rightarrow \infty }\Vert f-P_n\Vert _K^{\frac{1}{d_n}}\,&lt;\,1\]for every compact set $K\subset G_f$. If, moreover, the sequence $\lbrace d_{n+1}/d_n\rbrace $ is bounded then $G_f\,=\,{\mathbb{C}}^N$.If $E$ is a closed set in ${\mathbb{C}}^N$ then $V_E\,\equiv \, 0$ if and only if each series of homogeneous polynomials $\sum _{j=0}^{\infty }Q_j$, for which some subsequence $\lbrace s_{n_k}\rbrace $ of partial sums converges point-wise on $E$, possesses Ostrowski gaps relative to a subsequence $\lbrace n_{k_l}\rbrace $ of $\lbrace n_k\rbrace $.In one-dimensional setting these results are due to J. Müller and A. Yavrian [5].
LA - eng
KW - Taylor seies; Ostrowski gaps; Siciak extremal function
UR - http://eudml.org/doc/219851
ER -