A Convex Stone-Weierstrass Theorem & Applications

A convex-polynomial is a convex combination of the monomials $\{1, x, x^2, \ldots\}$. This paper establishes that the convex-polynomials on $\mathbb R$ are dense in $L^p(\mu)$ and weak$^*$ 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 $K \cap [-1, \infty) = \emptyset$, 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.