Local Korovkin-type approximation problems for bounded function spaces

The paper is concerned with some local approximation problems by special classes of positive linear operators acting on spaces of Borel measurable bounded functions on a compact metric space. A Korovkin-type criterion is established with respect to a given positive linear operator, the convergence being required uniformly on arbitrary but fixed compact subsets. Several special cases are discussed where the limit operator is a Markov projection, a composition operator as well as the identity operator. Some applications are illustrated which concern the Bernstein operator on the unit interval and on the multidimensional simplices as well as Bernstein–Schnabl operators on convex compact subsets of locally convex spaces together with their modifications associated with integrated generalized means. Finally, we highlight some local approximation properties of a sequence of positive linear operators which, among other things, generalize the Kantorovich operators to the settings of convex compact subsets of locally convex spaces.