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

Siciak, Józef

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Numéro Spécial : Actes du colloque Analyse Complexe et Applications en l’honneur de Nguyen Than Van, Tome 20 (2011) no. S2, pp. 189-209.

Soit $\Gamma$ un sous-ensemble non pluripolaire de ${ℂ}^N$. Soit $f$ une fonction holomorphe sur un voisinage ouvert connexe $G$ de $\Gamma$. Soit $\left\{P_n\right\}$ une suite de polynômes de degré $deg{P}_n\le d_n(d_n<d_{n+1})$ telle que

$$\underset{n\to \infty }{lim\; sup}{|f\left(z\right)-P_n\left(z\right)|}^{1/d_n}<1,z\in \Gamma .$$

On démontre que si

$$\underset{n\to \infty }{lim\; sup}{\left|P_n\left(z\right)\right|}^{1/d_n}\le 1,z\in E,$$

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

$$\underset{n\to \infty }{lim\; sup}{\parallel f-P_n\parallel }_K^{\frac{1}{d_n}}<1$$

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 $\left\{s_{n_k}\right\}$ de sommes partielles convergeant ponctuellement sur $E$, admet des lacunes de type Ostrowski relativement à une sous-suite $\left\{n_{k_l}\right\}$ de $\left\{n_k\right\}$.

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

Let $\Gamma$ be a non-pluripolar set in ${ℂ}^N$. Let $f$ be a function holomorphic in a connected open neighborhood $G$ of $\Gamma$. Let $\left\{P_n\right\}$ be a sequence of polynomials with $deg{P}_n\le d_n(d_n<d_{n+1})$ such that

$$\underset{n\to \infty }{lim\; sup}{|f\left(z\right)-P_n\left(z\right)|}^{1/d_n}<1,z\in \Gamma .$$

We show that if

$$\underset{n\to \infty }{lim\; sup}{\left|P_n\left(z\right)\right|}^{1/d_n}\le 1,z\in E,$$

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

$$\underset{n\to \infty }{lim\; sup}{\parallel f-P_n\parallel }_K^{\frac{1}{d_n}}<1$$

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 $\left\{s_{n_k}\right\}$ of partial sums converges point-wise on $E$, possesses Ostrowski gaps relative to a subsequence $\left\{n_{k_l}\right\}$ of $\left\{n_k\right\}$.

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

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