Properties of convergence for the q-Meyer-König and Zeller operators

Abstract

In this paper, we discuss properties of convergence for the q-Meyer-König and Zeller operators M n , q . Based on an explicit expression for M n , q ( t 2 , x ) in terms of q-hypergeometric series, we show that for q n ∈ ( 0 , 1 ] , the sequence ( M n , q n ( f ) ) n ⩾ 1 converges to f uniformly on [ 0 , 1 ] for each f ∈ C [ 0 , 1 ] if and only if lim n → ∞ q n = 1 . For fixed q ∈ ( 0 , 1 ) , we prove that the sequence ( M n , q ( f ) ) converges for each f ∈ C [ 0 , 1 ] and obtain the estimates for the rate of convergence of ( M n , q ( f ) ) by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. We also give explicit formulas of Voronovskaya type for the q-Meyer-König and Zeller operators for fixed 0 < q < 1 . If 0 < q < 1 , f ∈ C 1 [ 0 , 1 ] , we show that the rate of convergence for the Meyer-König and Zeller operators is o ( q n ) if and only if f ( 1 − q k − 1 ) − f ( 1 − q k ) ( 1 − q k − 1 ) − ( 1 − q k ) = f ′ ( 1 − q k ) , k = 1 , 2 , … .