Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Characterization of polynomials by L2-estimates

Abstract

The main result of this paper is that an entire function $f$ that is in $L^2({\mathbb{C}}^n,\psi )$ with respect to the weight $\psi (z)=2m{H}_S(z)+\gamma \log (1+|z{|}^2)$ is a polynomial with exponents in $m{\hat{S}}_{\Gamma }$. Here $H_S$ is the logarithmic supporting function of a compact convex set $S\subset {\mathbb{R}}_{+}^n$ with $0\in S$, $\gamma \ge 0$ is small enough in terms of $m$, and ${\hat{S}}_{\Gamma }$ is the hull of $S$ with respect to a certain cone $\Gamma$ depending on $S$, $m$ and $\gamma$. An example showing that in general ${\hat{S}}_{\Gamma }$ can not be replaced by $S$ is constructed.

7 pages, 2 figures