Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Siciak-Zakharyuta theorem

The classical Siciak-Zakharyuta theorem states that the Siciak-Zakharyuta function $V_E$ of a subset $E$ of ${\mathbb{C}}^n$, also called a pluricomplex Green function or global exremal function of $E$, equals the logarithm of the Siciak function ${\mathrm{\Phi }}_E$ if $E$ is compact. The Siciak-Zakharyuta function is defined as the upper envelope of functions in the Lelong class that are negative on $E$, and the Siciak function is the upper envelope of $m$-th roots of polynomials $p$ in ${\mathcal{P}}_m({\mathbb{C}}^n)$ of degree $\le m$ such that $\mathrm{\mid }p\mathrm{\mid }\le 1$ on $E$. We generalize the Siciak-Zakharyuta theorem to the case where the polynomial space ${\mathcal{P}}_m({\mathbb{C}}^n)$ is replaced by ${\mathcal{P}}_m^S({\mathbb{C}}^n)$ consisting of all polynomials with exponents restricted to sets $mS$, where $S$ is a compact convex subset of ${\mathbb{R}}_{+}^n$ with $0\in S$. It states that if $q$ is an admissible weight on a closed set $E$ in ${\mathbb{C}}^n$ then $V_{E,q}^S=\log {\mathrm{\Phi }}_{E,q}^S$ on ${\mathbb{C}}^{\ast n}$ if and only if the rational points in $S$ form a dense subset of $S$.