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 $m=0,1,2,3,\dots$, and uniform estimates of error in the approximation are replaced by weighted uniform estimates with respect to an admissible weight function.