Local Korovkin-type approximation problems for bounded function spaces (Q6552625)

Approximation by positive linear operators is a well-established and active area of research. In this context, the Korovkin-type approximation theory plays a prominent role. Its classical part is presented in [\textit{F. Altomare} and \textit{M. Campiti}, Korovkin-type approximation theory and its applications. Berlin: Walter de Gruyter (1994; Zbl 0924.41001); \textit{F. Altomare}, Surv. Approx. Theory 5, 92--164 (2010; Zbl 1285.41012)]. In [\textit{F. Altomare}, Expo. Math. 40, No. 4, 1229--1243 (2022; Zbl 1526.41007)] the author has initiated a new direction of research by establishing convergence criteria toward the identity operator for sequences of positive linear operators acting on spaces of Borel measurable bounded functions defined on a metric space. In this setting the convergence is uniform on compact subsets. The test sets involved in the main result are strongly related to the given distance on the metric space. \N\NThe paper contains several useful simple criteria for the local approximation of locally continuous Borel measurable bounded functions by positive linear operators. The results are presented both in one dimensional and multidimensional settings as well as in infinite dimensional settings (specifically, Hilbert spaces). Consequently, the paper enlarges the horizon of such lines of investigation by determining larger classes of test sets for local approximation problems. In particular, it turns out that every classical Korovkin subset in a space of continuous functions is a test set for the local approximation problems involving locally continuous Borel measurable bounded functions. An important new aspect is that the Korovkin-type convergence criteria are developed with respect to a given positive linear operator, not only the identity operator. This fact opens the way to present applications concerning Markov projections and composition operators, extending previous results obtained in literature. The methods used in this paper completely differ from the ones used in [\textit{F. Altomare}, Expo. Math. 40, No. 4, 1229--1243 (2022; Zbl 1526.41007)] and involve measure-theoretical tools. They are developed in the framework of compact metric spaces and for the special class of positive linear operators which admit an integral representation in terms of suitable Borel measures, leaving open the problem concerning non compact metric spaces and/or general sequences of positive linear operators. \N\NThe general results are accompanied by several applications which concern the iterates of Bernstein operators on the unit interval and on multidimensional simplices, Bernstein-Schnabl operators on metrizable convex compact subsets of locally convex spaces, their modification associated with integrated generalized means and, finally, a sequence of positive linear operators, introduced in [\textit{F. Altomare} et al., Banach J. Math. Anal. 11, No. 3, 591--614 (2017; Zbl 1383.41003)], which, among other things, generalize Kantorovich operators to the setting of metrizable convex compact subsets of locally convex spaces. The new results and the new methods of proof recommend this paper as a starting point for new developments.