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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 supn|f(z)-Pn(z)|1/dn<1,zΓ.

On démontre que si

lim supn|Pn(z)|1/dn1,zE,

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 supnf-PnK1dn<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 supn|f(z)-Pn(z)|1/dn<1,zΓ.

We show that if

lim supn|Pn(z)|1/dn1,zE,

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 supnf-PnK1dn<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].

@article{AFST_2011_6_20_S2_189_0,
     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, Toulouse},
     volume = {Ser. 6, 20},
     number = {S2},
     year = {2011},
     doi = {10.5802/afst.1312},
     zbl = {1229.32003},
     mrnumber = {2858174},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1312/}
}
Józef Siciak. 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. https://afst.centre-mersenne.org/articles/10.5802/afst.1312/

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