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Abstract

We apply pluripotential theory to establish results in $\mathbb R^k$ concerning uniform approximation by functions of the form $w^n P_n$ where $w$ denotes a continuous nonnegative function and $P_n$ is a polynomial of degree at most $n$. Then we use our work to show that on the intersection of compact sections ${\mit\Sigma} \subset \mathbb R^k$ a continuous function on ${\mit\Sigma}$ is uniformly approximable by $\theta$-incomplete polynomials (for a fixed $\theta,$ $0< \theta < 1$) iff $f$ vanishes on $\theta^2 {\mit\Sigma}$. The class of sets ${\mit\Sigma}$ expressible as the intersection of compact sections includes the intersection of a symmetric convex compact set with a single orthant.