Probabilistic methods in the approximation by linear positive operators

Abstract

By the theorem of Daniell-Stone a probabilistic interpretation of sequences of linear positive operators in approximation theory is given which yields a generalization of Korovkin's theorem in the context of uniform integrability. For a class of sequences of linear positive operators defined by using random variables asymptotic results on the goodness of local and global approximation are proved, where in the latter case an asymptotically best constant in connection with the modulus of continuity is determined. There are used central limit theorems, especially a generalization of the Berry-Esseen theorem due to Osipov-Petrov (1967) and Feller (1968).