A Convex Stone-Weierstrass Theorem & Applications

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A convex-polynomial is a convex combination of the monomials {1, x, x², }. This paper establishes that the convex-polynomials on R are dense in Lᵖ(μ) and weak* dense in L∞(μ), precisely when μ([-1,∞)) = 0. It is shown that the convex-polynomials are dense in C(K) precisely when K ∩ [-1, ∞) =, 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 ∈ 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.