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 $\mathbb{C}^{*n}$ 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 H\"ormander's $L^2$-theory and Siciak-Zakharyuta functions $V^S_K$ associated to $S$. We provide a formula for $V^S_K$ when $K$ is a rotationally symmetric compact subset of $\mathbb{C}^{*n}$.