Polynomials with exponents in compact convex sets and associated weighted extremal functions – The Bernstein–Walsh–Siciak theorem

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 $m=0,1,2,\dots,$ and uniform estimates of error in the approximation are replaced by weighted uniform estimates with respect to an admissible weight function.