Robin functions and extremal functions

• Autorzy: T. Bloom , N. Levenberg , S. Ma'u
• Języki publikacji: EN

Abstrakty

EN

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, sup{1/(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.

Kategorie tematyczne

• 31C15: Potentials and capacities

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2003
• Tom: 80
• Numer: 1
• Strony: 55-84

Daty

wydano

2003

Twórcy

autor

T. Bloom

• Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 3G3

autor

N. Levenberg

• University of Auckland, Private Bag 92019, Auckland, New Zealand

autor

S. Ma'u

• University of Auckland, Private Bag 92019, Auckland, New Zealand

Identyfikatory

DOI

10.4064/ap80-0-4