Robin functions and extremal functions

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

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

Annales Polonici Mathematici, Tome 81 (2003), p. 55-84 / Harvested from The Polish Digital Mathematics Library

Résumé

Given a compact set $K \subset ℂ^{N}$ , 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, ${| | p | |}_{K} \leq 1$ ]. Our main result is that if K is regular, then all of the functions $V_{K}^{(n)}$ are continuous; and their associated Robin functions $ϱ_{V_{K}^{(n)}} (z) : = l i m s u p_{| λ | \to \infty} [V_{K}^{(n)} (λ z) - l o g (| λ |)]$ increase to $ϱ_{K} : = ϱ_{V_{K}}$ for all z outside a pluripolar set.

Publié le : 2003-01-01

Zbl 1031.31004

EUDML-ID : urn:eudml:doc:286634

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     title = {Robin functions and extremal functions},
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T. Bloom; N. Levenberg; S. Ma'u. Robin functions and extremal functions. Annales Polonici Mathematici, Tome 81 (2003) pp. 55-84. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap80-0-4/