A Convex Stone-Weierstrass Theorem & Applications

Abstract

A convex-polynomial is a convex combination of the monomials $\{1,x,x^2,\dots \}$. This paper establishes that the convex-polynomials on $\mathbb{R}$ are dense in $L^p(\mu )$ and weak${}^{\ast }$ dense in $L^{\infty }(\mu )$, precisely when $\mu ([-1,\infty ))=0$. It is shown that the convex-polynomials are dense in $C(K)$ precisely when $K\cap [-1,\infty )=\varnothing$, where $K$ is a compact subset of the real line. Moreover, the closure of the convex-polynomials on $[-1,b]$ are shown to be the functions that have a convex-power series representation. A continuous linear operator $T$ on a locally convex space $X$ is convex-cyclic if there is a vector $x\in X$ such that the convex hull of the orbit of $x$ is dense in $X$. The above results characterize which multiplication operators on various real Banach spaces are convex-cyclic. It is shown for certain multiplication operators that every closed invariant convex set is a closed invariant subspace.