Sets in 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 (2011)

  • Volume: 20, Issue: S2, page 189-209
  • ISSN: 0240-2963

Abstract

top
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 deg P 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 K G 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].

How to cite

top

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 -

References

top
  1. Błocki (Z.).— Equilibrium measure of a product subset of n , PAMS, 128(12), p. 3595-3599 (2000). Zbl0959.32039MR1707508
  2. Cegrell (U.), Kołodziej (S.) and Levenberg (N.).— Two problems on potential theory for unbounded sets, p. 265-276, Math. Scand., 83 (1998). Zbl0928.31001MR1673930
  3. Hayman (W. K.).— Subharmonic Functions, Vol. 2 Academic Press (1989). Zbl0419.31001MR1049148
  4. Klimek (M.).— Pluripotential Theory Oxford Univ. Press (1991). Zbl0742.31001MR1150978
  5. Müller (J.) and Yavria (A.).— On polynomial sequences with restricted growth near infinity, Bull. London Math. Soc., 34, p. 189-199 (2002). Zbl1020.30044MR1874246
  6. Siciak (J.).— Extremal plurisubharmonic functions in n , Ann. Polon. Math., 39, p. 175-211 (1981). Zbl0477.32018MR617459
  7. Siciak (J.).— Extremal plurisubharmonic functions and capacities in n , Sophia Kokyuroku in Mathematics, 14 Sophia University, Tokyo (1982). Zbl0579.32025
  8. BTaylor (B.A.).— An estimate for an extremal plurisubharmonic function in n , Seminaire P. Lelong, P. Dolbeault, H. Skoda (Analyse), Lecture Notes in Math., 1028, Springer Verlag, 318-328 (1983). Zbl0522.32014
  9. Truong Tuyen Trung.— Sets non-thin at in m , J. Math. Anal. Appl., 356(2), p. 517-524 (2009). Zbl1168.32001MR2524286

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.