A Choquet-Deny-Type theorem and applications to approximation in weighted spaces (Q2504865)

A weight on a locally compact space Hausdorff \(X\) is a non-negative upper semi-continuous function \(v:X\to\mathbb R.\) For a directed set \(V\) of weights one denotes by \(CV_\infty(X)\) the space of all real-valued continuous functions \(f\) on \(X\) such that \(v\cdot f\) vanishes at infinity for every \(v\in V\). A weighted space is a space \(CV_\infty(X)\) equipped with the locally convex topology \(\omega_V\) generated by the family of seminorms \(p_v(f)=\sup\{v(x)|f(x)| : x\in X\},\, f\in CV_\infty(X),\, v\in V\) (see \textit{L. Nachbin} [Elements of Approximation Theory, Van Nostrand, Princeton, N.J. (1967; Zbl 0173.41403); reprint Huntington, N.Y. (1976; Zbl 0331.46015)]). The authors extend to this framework the result of \textit{G. Choquet} and \textit{J. Deny} [J. Math. Pures Appl. (9) 36, 179--189 (1957; Zbl 0077.31402)], on the characterization of convex cones \(S\subset CV_\infty(X)\) which are inf-lattices. For instance, \(f\in \overline S\) if and only if for every monotone \(\omega_V\)-continuous affine functional \(\psi\) on \(CV_\infty(X)\) and \(x\in X\) such that \(\psi(f)> f(x)\) there exists \(g\in S\) such that \(\psi(g)> g(x)\) (Theorem 2.1). As applications one gives characterizations of Korovkin closure of vector lattices and one proves a Stone--Weierstrass type theorem for weighted spaces.