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

Leokadia Bialas-Ciez

Annales Polonici Mathematici, Tome 105 (2012), p. 41-51 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $ℂ^{N}$.

Détail

Publié le : 2012-01-01

Zbl 1254.32048

EUDML-ID : urn:eudml:doc:286610