On Convergence Properties of Szasz-Mirakyan-Bernstein Operators of two Variables

Tuncay Tunc

doi:10.13189/ms.2013.010309

Abstract

In this study, we have constructed a sequence of new positive linear operators with two vari- able by using Szasz-Mirakyan and Bernstein Operators, and investigated its approximation properties.

Highlights

Let n ∈ N = {1, 2, ...} and f ∈ C[0, 1]
The Bernstein polynomials are used for important applications in the branches of mathematics, for example, approximation theory, probability theory, number theory, the solution of the integral and differential equations and the others (e.g. [7, 1, 6, 9])
For y ∈ [0, ∞) and f ∈ C(D), let us define the function fy ∈ C[0, 1] by fy(x) := f (x, y)

Summary

Introduction

For y ∈ [0, ∞) and f ∈ C(D), let us define the function fy ∈ C[0, 1] by fy(x) := f (x, y) With this notation, the positive linear operators Ln given by (5) can be written in the form. Some positive linear operators for the functions of two variables are introduced and investigated their approximation properties by the authors in [13, 5, 11, 15, 14, 12, 10, 4]. Taking into account the Bernstein polynomials and the Szasz-Mirakyan operators, we introduce some positive linear operators for functions of two variables

Some Notations and Auxiliary Facts

Partial Derivatives