Siciak’s extremal function via Bernstein and Markov constants for compact sets in ${ℂ}^N$

Leokadia Bialas-Ciez. "Siciak’s extremal function via Bernstein and Markov constants for compact sets in $ℂ^{N}$." Annales Polonici Mathematici 106.1 (2012): 41-51. <http://eudml.org/doc/286610>.

@article{LeokadiaBialas2012,
abstract = {The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E ⊂ ℂ^\{N\}$. We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function $Φ_\{E\}$. Moreover, we show that one of these extremal-like functions is equal to $Φ_\{E\}$ if E is a nonpluripolar set with $lim_\{n→∞\} Mₙ(E)^\{1/n\} = 1$ where $Mₙ(E) := sup\{|| |grad P| ||_\{E\}/||P||_\{E\}\}$, the supremum is taken over all polynomials P of N variables of total degree at most n and $||·||_\{E\}$ is the uniform norm on E. The above condition is fulfilled e.g. for all regular (in the sense of the continuity of the pluricomplex Green function) compact sets in $ℂ^\{N\}$.},
author = {Leokadia Bialas-Ciez},
journal = {Annales Polonici Mathematici},
keywords = {Siciak's extremal function; Bernstein inequality; Markov inequality; plurisubharmonic functions; pluricomplex Green function},
language = {eng},
number = {1},
pages = {41-51},
title = {Siciak’s extremal function via Bernstein and Markov constants for compact sets in $ℂ^\{N\}$},
url = {http://eudml.org/doc/286610},
volume = {106},
year = {2012},
}

TY - JOUR
AU - Leokadia Bialas-Ciez
TI - Siciak’s extremal function via Bernstein and Markov constants for compact sets in $ℂ^{N}$
JO - Annales Polonici Mathematici
PY - 2012
VL - 106
IS - 1
SP - 41
EP - 51
AB - The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E ⊂ ℂ^{N}$. We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function $Φ_{E}$. Moreover, we show that one of these extremal-like functions is equal to $Φ_{E}$ if E is a nonpluripolar set with $lim_{n→∞} Mₙ(E)^{1/n} = 1$ where $Mₙ(E) := sup{|| |grad P| ||_{E}/||P||_{E}}$, the supremum is taken over all polynomials P of N variables of total degree at most n and $||·||_{E}$ is the uniform norm on E. The above condition is fulfilled e.g. for all regular (in the sense of the continuity of the pluricomplex Green function) compact sets in $ℂ^{N}$.
LA - eng
KW - Siciak's extremal function; Bernstein inequality; Markov inequality; plurisubharmonic functions; pluricomplex Green function
UR - http://eudml.org/doc/286610
ER -