Sets in N with vanishing global extremal function and polynomial approximation
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. S2, pp. 189-209.

Soit Γ un sous-ensemble non pluripolaire de N . Soit f une fonction holomorphe sur un voisinage ouvert connexe G de Γ. Soit {P n } une suite de polynômes de degré degP n d n (d n <d n+1 ) telle que

lim sup n | f ( z ) - P n ( z ) | 1 / d n < 1 , z Γ .

On démontre que si

lim sup n | P n ( z ) | 1 / d n 1 , z E ,

E is est un sous-ensemble de N tel que la fonction extrémale globale V E 0 sur N , alors le domaine maximal d’existence G f de f est uniforme, et

lim sup n f - P n K 1 d n < 1

pour tout compact KG f . Si, de plus, la suite {d n+1 /d n } est bornée alors G f = N .

Si E est un sous-ensemble fermé de N alors V E 0 si et seulement si chaque série de polynômes homogènes j=0 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].

Let Γ be a non-pluripolar set in N . Let f be a function holomorphic in a connected open neighborhood G of Γ. Let {P n } be a sequence of polynomials with degP n d n (d n <d n+1 ) such that

lim sup n | f ( z ) - P n ( z ) | 1 / d n < 1 , z Γ .

We show that if

lim sup n | P n ( z ) | 1 / d n 1 , z E ,

where E is a set in N such that the global extremal function V E 0 in N , then the maximal domain of existence G f of f is one-sheeted, and

lim sup n f - P n K 1 d n < 1

for every compact set KG f . If, moreover, the sequence {d n+1 /d n } is bounded then G f = N .

If E is a closed set in N then V E 0 if and only if each series of homogeneous polynomials j=0 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].

DOI : 10.5802/afst.1312
Siciak, Józef 1

1 Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
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Siciak, Józef. Sets in ${\mathbb{C}}^N$ with vanishing global extremal function and polynomial approximation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. S2, pp. 189-209. doi : 10.5802/afst.1312. http://archive.numdam.org/articles/10.5802/afst.1312/

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