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