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We generalize the Bernstein-Walsh-Siciak theorem on polynomial approximationin $\mathbb{C}^n$ 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 exponentsrestricted 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 uniformestimates of error in the approximation are replaced by weighted uniformestimates with respect to an addmissible weight function.

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