Some approximation results on Bernstein-Schurer operators defined by ( p , q ) $(p,q)$ -integers

Mohammad Mursaleen,Ashirbayev Nurgali,Md Nasiruzzaman

doi:10.1186/s13660-015-0767-4

Abstract

Recently, Mursaleen et al. (On $(p,q)$ -analogue of Bernstein operators, arXiv:1503.07404 ) introduced and studied the $(p,q)$ -analog of Bernstein operators by using the idea of $(p,q)$ -integers. In this paper, we generalize the q-Bernstein-Schurer operators using $(p,q)$ -integers and obtain a Korovkin type approximation theorem. Furthermore, we obtain the convergence of the operators by using the modulus of continuity and prove some direct theorems.

Highlights

Introduction and preliminariesIn, Bernstein [ ] introduced the following sequence of operators Bn : C[, ] → C[, ] defined for any n ∈ N and f ∈ C[, ]: n Bn(f ; x) =n xk( – x)n–kf k, x ∈ [, ]. k n ( . ) k=By applying the idea of q-integers, the q-Bernstein operators were introduced by Lupaş [ ] and later by Philip [ ]
We investigate some approximation properties of these operators and obtain the rate of convergence by using the modulus of continuity
Proof The proof is based on the well-known Korovkin theorem regarding the convergence of a sequence of linear and positive operators, so it is enough to prove the conditions

Summary

For q

In order to obtain the convergence results of the operator Bpm,q, , we take a sequence qm ∈ ( , ) and pm ∈ Such that limm→∞ pm = and limm→∞ qm = , so we get limm→∞[m]pm,qm = ∞. Proof The proof is based on the well-known Korovkin theorem regarding the convergence of a sequence of linear and positive operators, so it is enough to prove the conditions. (e ; By making a simple calculation we get lim [m + ]pm,qm = , m→∞ [m]pm,qm as < qm < pm ≤.

Then there exists a positive constant

On the other hand we have

Conclusion