The convergence of $(p,q)$-Bernstein operators for the Cauchy kernel with a pole via divided difference

Abstract

In this paper, some qualitative approximation results for the (p, q)-Bernstein operators B n p,q (f ; x) are obtained for the Cauchy kernel 1 x-α with a pole α ∈ [0, 1] for q > p > 1. The main focus lies in the study of behavior of operators B n p,q (f ; x) for the function f m (x) = 1 x-p m q-m , x = p m q-m and f m (p m q-m) = a, a ∈ R and the extra parameter p provides flexibility for the approximation.