On the order of approximation of continuous functions by positive linear operators of finite rank

Abstract

Let { L n } nϵ N be a sequence of positive linear operators from C[0, 1] into B(Ω), where B(Ω) is the space of real bounded functions over Ω ⊂ [0, 1], meas (Ω) > 0. Suppose that for each n the linear space {L nƒ:ƒ ϵ C[0, 1]} has dimension n + 1. It is shown that the quantity n 2 ∑ j=0 2 ƒL n(t j;x)−x jƒ does not tend to zero on a set of positive measure.