Positive linear operators and summability

Abstract

Let { L n } be a sequence of positive linear operators defined on C [ a, b ] of the form where x nk ∈ [ a, b ] for each k = 0, 1,…, n = 1, 2,…. The convergence properties of the sequences { L n ( f )} to for each f ∈ C [ a, b ] have been the object of much recent research (see e.g. [4], [8], [11], [13]). In many cases positive linear operators of the form (1) give rise to interesting summability matrices A = ( a nk (x)) and vice- versa.