A Green's function for $\theta$-incomplete polynomials

Joe Callaghan¹

¹ Department of Mathematics University of Toronto Toronto, Ontario M5S 2E4, Canada

Annales Polonici Mathematici, Tome 90 (2007) no. 1, pp. 21-35

Résumé

Let $K$ be any subset of $ \mathbb C^N $. We define a pluricomplex Green's function $V_{K,\theta} $ for $ \theta $-incomplete polynomials. We establish properties of $V_{K,\theta} $ analogous to those of the weighted pluricomplex Green's function. When $K$ is a regular compact subset of $ \mathbb R^N $, we show that every continuous function that can be approximated uniformly on $K$ by $\theta$-incomplete polynomials, must vanish on $ K \setminus {\rm supp}\,(dd^{c} V_{K,\theta})^N $. We prove a version of Siciak's theorem and a comparison theorem for $ \theta $-incomplete polynomials. We compute $ {\rm supp}\,(dd^{c} V_{K,\theta})^N $ when $K$ is a compact section.

Détail

DOI : 10.4064/ap90-1-2

Keywords: subset mathbb define pluricomplex greens function theta theta incomplete polynomials establish properties theta analogous those weighted pluricomplex greens function regular compact subset mathbb every continuous function approximated uniformly theta incomplete polynomials vanish setminus supp theta prove version siciaks theorem comparison theorem theta incomplete polynomials compute supp theta compact section