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

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 $\psi (z)=2m{H}_S(z)+\gamma \log (1+\mathrm{\mid }z{\mathrm{\mid }}^2)$ is a polynomial with exponents in $m{\widehat{S}}_{\mathrm{\Gamma }}$. 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 ${\widehat{S}}_{\mathrm{\Gamma }}$ is the hull of $S$ with respect to a certain cone $\mathrm{\Gamma }$ depending on $S$, $m$ and $\gamma$. An example showing that in general ${\widehat{S}}_{\mathrm{\Gamma }}$ can not be replaced by $S$ is constructed.