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 $\psi(z)=2mH_S(z)+\gamma\log(1+|z|^2)$ is a polynomial with exponents in $m\widehat S_\Gamma$. Here $H_S$ is the logarithmic supporting function of a compact convex set $S\subset \mathbb R^n_+$ with $0\in S$, $\gamma\geq 0$ is small enough in terms of $m$, and $\widehat S_\Gamma$ is the hull of $S$ with respect to a certain cone $\Gamma$ depending on $S$, $m$ and $\gamma$. An example showing that in general $\widehat S_\Gamma$ can not be replaced by $S$ is constructed.