Robin functions and extremal functions

T. Bloom; N. Levenberg; S. Ma'u

Annales Polonici Mathematici (2003)

  • Volume: 80, Issue: 1, page 55-84
  • ISSN: 0066-2216

Abstract

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Given a compact set K ⊂ ℂ N , for each positive integer n, let V ( n ) ( z ) = V K ( n ) ( z ) := sup 1 / ( d e g p ) V p ( K ) ( p ( z ) ) : p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions V K ( n ) are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, V K ( z ) := max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, | | p | | K ≤ 1 ]. Our main result is that if K is regular, then all of the functions V K ( n ) are continuous; and their associated Robin functions ϱ V K ( n ) ( z ) : = l i m s u p | λ | → ∞ [ V K ( n ) ( λ z ) - l o g ( | λ | ) ] increase to ϱ K : = ϱ V K for all z outside a pluripolar set.

How to cite

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T. Bloom, N. Levenberg, and S. Ma'u. "Robin functions and extremal functions." Annales Polonici Mathematici 80.1 (2003): 55-84. <http://eudml.org/doc/286634>.

@article{T2003,
abstract = {Given a compact set $K ⊂ ℂ^\{N\}$, for each positive integer n, let $V^\{(n)\}(z)$ = $V^\{(n)\}_\{K\}(z)$ := sup$1/(deg p) V_\{p(K)\}(p(z))$: p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions $V^\{(n)\}_\{K\}$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $V_\{K\}(z)$:= max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, $||p||_\{K\} ≤ 1$]. Our main result is that if K is regular, then all of the functions $V^\{(n)\}_\{K\}$ are continuous; and their associated Robin functions $ϱ_\{V^\{(n)\}_\{K\}\}(z) := limsup_\{|λ|→∞\} [V^\{(n)\}_\{K\}(λz) - log(|λ|)]$ increase to $ϱ_\{K\} := ϱ_\{V_\{K\}\}$ for all z outside a pluripolar set.},
author = {T. Bloom, N. Levenberg, S. Ma'u},
journal = {Annales Polonici Mathematici},
keywords = {plurisubharmonic function; extremal function; Robin function},
language = {eng},
number = {1},
pages = {55-84},
title = {Robin functions and extremal functions},
url = {http://eudml.org/doc/286634},
volume = {80},
year = {2003},
}

TY - JOUR
AU - T. Bloom
AU - N. Levenberg
AU - S. Ma'u
TI - Robin functions and extremal functions
JO - Annales Polonici Mathematici
PY - 2003
VL - 80
IS - 1
SP - 55
EP - 84
AB - Given a compact set $K ⊂ ℂ^{N}$, for each positive integer n, let $V^{(n)}(z)$ = $V^{(n)}_{K}(z)$ := sup$1/(deg p) V_{p(K)}(p(z))$: p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions $V^{(n)}_{K}$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $V_{K}(z)$:= max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, $||p||_{K} ≤ 1$]. Our main result is that if K is regular, then all of the functions $V^{(n)}_{K}$ are continuous; and their associated Robin functions $ϱ_{V^{(n)}_{K}}(z) := limsup_{|λ|→∞} [V^{(n)}_{K}(λz) - log(|λ|)]$ increase to $ϱ_{K} := ϱ_{V_{K}}$ for all z outside a pluripolar set.
LA - eng
KW - plurisubharmonic function; extremal function; Robin function
UR - http://eudml.org/doc/286634
ER -

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