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

Abstract

Abstract In this paper, some qualitative approximation results for the (p,q) $(p,q)$-Bernstein operators Bp,qn(f;x) $B_{p,q}^{n}(f;x)$ are obtained for the Cauchy kernel 1x−α $\frac{1}{x-\alpha }$ with a pole α∈[0,1] $\alpha \in {}[ 0,1]$ for q>p>1 $q>p>1$. The main focus lies in the study of behavior of operators Bp,qn(f;x) $B_{p,q}^{n}(f;x)$ for the function fm(x)=1x−pmq−m $f_{m}(x)=\frac{1}{x-p^{m}q^{-m}}$, x≠pmq−m $x\neq p^{m}q^{-m}$ and fm(pmq−m)=a $f_{m}(p^{m}q ^{-m})=a$, a∈R $a\in \mathbb{R}$ and the extra parameter p provides flexibility for the approximation.

Keywords

• ( p , q ) $(p,q)$ -integer
• ( p , q ) $(p,q)$ -Bernstein operator
• Convergence
• Approximation of unbounded function
• Cauchy kernel