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

Abstract

For a certain class of sequences ( L n ) of linear positive operators that are related to sums of independent identically distributed random variables U kx , the sequence ( D n ) defined by D n(L n,B):= sup{|L nf−f|ω(f,n − 1 2 ) −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 ( D n ) is proved to be O( h(√ n) −1) where h is an appropriate function such that the expectation of (U 1x−EU 1x) 2h(U 1x−EU 1x) is bounded. By a more detailed analysis of the sequence ( D n( M ̃ n,[0,1]) ) for the special case of the operators M ̃ n of Meyer-König and Zeller it is shown that sup> {D n( M ̃ n, [0,1]): n∈ N}= 37−16√3 9 .