Korovkin-type theorems and local approximation problems

Abstract

Of concern are local approximation problems for sequences of positive linear operators acting on linear subspaces of functions defined on a metric space. A Korovkin-type theorem is established in such a framework together with several consequences related to one dimensional, multidimensional and infinite dimensional settings (Hilbert spaces).Furthermore, some applications are discussed which concern classical sequences of positive linear operators including (one dimensional and multidimensional) Bernstein operators, Kantorovich operators, Szász–Mirakyan operators, Gauss–Weierstrass operators and Bernstein–Schnabl operators on convex subsets of Hilbert spaces.Finally the paper ends with a reassessment of a result of Korovkin concerning subspaces of bounded 2π− periodic functions on R and with an application related to sequences of convolution operators generated by positive approximate identities.