Approximation Theorem for New Modification of q -Bernstein Operators on (0,1)

Abstract

In this work, we extend the works of F. Usta and construct new modified $q$ -Bernstein operators using the second central moment of the $q$ -Bernstein operators defined by G. M. Phillips. The moments and central moment computation formulas and their quantitative properties are discussed. Also, the Korovkin-type approximation theorem of these operators and the Voronovskaja-type asymptotic formula are investigated. Then, two local approximation theorems using Peetre’s $\mathbf{K}$ -functional and Steklov mean and in terms of modulus of smoothness are obtained. Finally, the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators are studied, and some graphs and numerical examples are shown by using Matlab algorithms.