On the convergence of sequences of positive linear operators and functionals on bounded function spaces

Abstract

Of concern are some simple criteria about the convergence of sequences of positive linear operators and functionals in the framework of spaces of bounded functions which are continuous on a given subset of their domain. Among other things some applications concerning the behaviour of the iterates of Bernstein operators defined both on [ 0 , 1 ] [0,1] and on the d d -dimensional simplex and hypercube ( d ≥ 1 ) (d\geq 1) are discussed. A final section treats the behaviour of integrated arithmetic means with respect to a probability Borel measure on a convex compact subset K K of a locally convex space. As a consequence a general weak law of large numbers for sequences of K − K- valued random variables is derived.