Statistical approximation of modified Schurer-type q-Bernstein Kantorovich operators

Qiu Lin

doi:10.1186/1029-242x-2014-465

Abstract

New modified Schurer-type q-Bernstein Kantorovich operators are introduced. The local theorem and statistical Korovkin-type approximation properties of these operators are investigated. Furthermore, the rate of approximation is examined in terms of the modulus of continuity and the elements of Lipschitz class functions. MSC:41A10, 41A25, 41A36.

Highlights

In, Lupaş [ ] introduced a q-analogue of Bernstein operators, and in another qgeneralization of the Bernstein polynomials was introduced by Phillips [ ]
After that generalizations of the Bernstein polynomials based on the q-integers attracted a lot of interest and were studied widely by a number of authors
The study of the statistical convergence for sequences of positive operators was attempted by Gadjiev and Orhan [ ]

Summary

Introduction

In , Lupaş [ ] introduced a q-analogue of Bernstein operators, and in another qgeneralization of the Bernstein polynomials was introduced by Phillips [ ]. Some new generalizations of well-known positive linear operators based on q-integers were introduced and studied by several authors (e.g., see [ – ]). The study of the statistical convergence for sequences of positive operators was attempted by Gadjiev and Orhan [ ]. In Section we study local convergence properties in terms of the first and the second modulus of continuity.

Results

Conclusion