  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 $\Phi_E$ 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 $\leq m$ such that
$|p|\leq 1$ 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 $V^S_{E,q}=\log\Phi^S_{E,q}$ on $\mathbb C^{*n}$
if and only if the rational points in $S$ form a dense subset of $S$.
