On the degree of approximation of continuous functions by a class of sequences of linear positive operators

AbstractFor a certain class of sequences (Ln) of linear positive operators that are related to sums of independent identically distributed random variables Ukx, the sequence (Dn) defined by Dn(Ln,B):=sup{|Lnf−f|ω(f,n−12)−1: f∈C(B)} where ‖.‖ denotes Čebyšev-norm and B a suitable compact interval in R, is known to be convergent. The rate of convergence of (Dn) is proved to be O(h(√n)−1) where h is an appropriate function such that the expectation of (U1x−EU1x)2h(U1x−EU1x) is bounded. By a more detailed analysis of the sequence (Dn(M̃n,[0,1])) for the special case of the operators M̃n of Meyer-König and Zeller it is shown that sup> {Dn(M̃n, [0,1]): n∈N}=37−16√39