Holomorphic approximation by polynomials with exponents restricted to a convex cone

We study the approximation of holomorphic functions of several complex variables by the ring ${\mathcal{P}}^S({\mathbb{C}}^n)$ of polynomials whose exponents are restricted to a convex cone ${\mathbb{R}}_{+}S$ for some compact convex $S\in {\mathbb{R}}_{+}^n$. We show a version of the Runge-Oka-Weil Theorem on approximation by these subrings on compact subsets of ${\mathbb{C}}^{\ast n}$ that are convex with respect to ${\mathcal{P}}^S({\mathbb{C}}^n)$. We show a sharper result on rotationally symmetric compact sets. The tools used are Hörmander’s $L^2$-theory and Siciak-Zakharyuta functions $V_K^S$ associated to $S$. We provide a formula for $V_K^S$ when $K$ is a rotationally symmetric compact subset of ${\mathbb{C}}^{\ast n}$.