A Kallin's lemma for polynomials with exponents in a compact convex set and local approximation

Abstract

Let $S\subset\Bbb R_+^n$ be compact and convex with $0\in S$, and let $\mathcal{P}^S$ be the algebra spanned by monomials with exponents in $\bigcup_{m\ge0}(mS\cap\Bbb N_0^n)$. We define the associated polynomial hull and prove a Kallin lemma for the union of two sets convex with respect to this algebra. A weighted approximation lemma then shows that, under a sector or ray condition, two admissible monomials generate exactly the functions of the unrestricted algebra that are constant on a specified zero fibre. Applications include an exact density criterion for mixed monomials, a transfer principle for relative perturbations, and density and hull computations on stars of totally real discs.