Contractive Korovkin subsets in weighted spaces of continuous functions
Weighted spaces \(CV_0(X)\) of continuous functions were introduced in the 1960s in connection with approximation theory and the weighted approximation problem [see \textit{L. Nachbin}, ``Elements of Approximation Theory'', van Nostrand Math. Studies 14 (1967; Zbl 0173.41403)]. They yield a general framework which includes virtually all locally convex spaces of continuous functions which occur in the applications. More specifically, let \(X\) denote a completely regular Hausdorff space and \(V\) a system of nonnegative upper semicontinuous functions on \(X\) (which can be assumed to be directed upward in the sense that for \(v_1, v_2 \in V\) and \(\lambda > 0\), there exists \(v_3 \in V\) with \(\lambda v_1, \lambda v_2 \leq v_3\); such systems have been called Nachbin families). Then \(CV_0(X)\) is the locally convex space of all continuous (real- or complex-valued) functions \(f\) for which \(vf\) vanishes at infinity on \(X\) for each \(v \in V\), endowed with the system \((p_v)_{v \in V}\) of seminorms given by \[ p_v(f) =\sup_{x \in X} v(x)|f(x)|\;\text{for} f \in CV_0(X). \] \textit{W. Roth} [Bull. Aust. Math. Soc. 55, 239-248 (1997; Zbl 0896.46016)] characterized the Korovkin closures of subspaces of \(CV_0(X)\) with respect to positive linear operators (for \(X\) locally compact). The present authors are interested in similar characterizations when positive linear contractions take the place of positive linear operators. Throughout the article under review they only consider the case that the system \(V\) consists of one single weight \(v\). Then \(CV_0(X)\) is an AM-space, and in this abstract framework characterizations of contractive Korovkin closures are already known [see \textit{H. O. Flösser} [J. Approximation Theory 31, 118-137 (1981; Zbl 0496.41024)]. However, the authors here prefer to follow a direct approach. They also study a similar problem when the operators take their values in the intersection of \(CV_0(X)\) with the space of bounded functions, and establish criteria to recognize extended Korovkin subspaces (provided \(v\leq 1\)).
scientific article; zbMATH DE number 1884332