  We generalize the Bernstein-Walsh-Siciak theorem on polynomial approximation
in $\mathbb{C}^n$ to the case where the polynomial ring
$\mathcal{P}(\mathbb{C}^n)$ is replaced by a subring
$\mathcal{P}^S(\mathbb{C}^n)$ consisting of all polynomials with exponents
restricted to sets $mS$, where $S$ is a compact convex subset of
$\mathbb{R}_+^n$ with $0 \in S$ and $m = 0, 1, 2, 3, \dots$, and uniform
estimates of error in the approximation are replaced by weighted uniform
estimates with respect to an admissible weight function.
