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2019 Convex Stone-Weierstrass theorems and invariant convex sets
Nathan S. Feldman, Paul J. McGuire
Rocky Mountain J. Math. 49(8): 2587-2611 (2019). DOI: 10.1216/RMJ-2019-49-8-2587

Abstract

A \textit {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 $L^\infty (\mu )$ whenever $\mu $ is a compactly supported regular Borel measure on $\mathbb {R}$ and $\mu ([-1,\infty )) = 0$. It is also shown that the convex polynomials are norm 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 $[-1,b]$ is shown to be the functions that have a convex power series representation.

A continuous linear operator $T$ on a locally convex space $X$ is \textit {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 previous results are used to characterize which multiplication operators on various real Banach spaces are convex-cyclic. Also, it is shown for certain multiplication operators that every nonempty closed invariant convex set is a closed invariant subspace.

Citation

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Nathan S. Feldman. Paul J. McGuire. "Convex Stone-Weierstrass theorems and invariant convex sets." Rocky Mountain J. Math. 49 (8) 2587 - 2611, 2019. https://doi.org/10.1216/RMJ-2019-49-8-2587

Information

Published: 2019
First available in Project Euclid: 31 January 2020

zbMATH: 07163188
MathSciNet: MR4058339
Digital Object Identifier: 10.1216/RMJ-2019-49-8-2587

Subjects:
Primary: 41A10
Secondary: ‎46E15, 47A16

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 8 • 2019
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