Advances in Mathematical Physics

Volume 2018, Article ID 3658389, 10 pages

https://doi.org/10.1155/2018/3658389

## The Approximation of Bivariate Blending Variant Szász Operators Based Brenke Type Polynomials

Department of Mathematics, University of Prishtina, Mother Teresa, 10000 Prishtina, Kosovo

Correspondence should be addressed to Artan Berisha; ude.rp-inu@ahsireb.natra

Received 13 July 2018; Accepted 3 October 2018; Published 2 December 2018

Academic Editor: Ivan Giorgio

Copyright © 2018 Behar Baxhaku et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We have constructed a new sequence of positive linear operators with two variables by using Szasz-Kantorovich-Chlodowsky operators and Brenke polynomials. We give some inequalities for the operators by means of partial and full modulus of continuity and obtain a Lipschitz type theorem. Furthermore, we study the convergence of Szasz-Kantorovich-Chlodowsky-Brenke operators in weighted space of function with two variables and estimate the rate of approximation in terms of the weighted modulus of continuity.

#### 1. Introduction

The well-known Mirakjan-Favard-Szasz type operators of one variable are defined aswhereand is such that the above exist series. For the convergence of to , usually is supposed to be the exponential growth, that is, , for all , with (see [1]). Later, in 1969, Jakimovski and Leviatan [2] investigated approximation properties of the generalization of Szász operators by means of the Appell polynomials which satisfy the identity where and Varma and Tasdelen [3] constructed positive linear operators based on orthogonal polynomials, e.g., Brenke polynomials. Suppose that is analytic functions in the disk , where and are real. The generating function for these polynomials is given by from which the explicit form of is as follows:

We suppose that(1), ,(2),(3)(5) and (6) converge for ,(4), for

Atakut and Buyukyazici in [4] introduced the Kantorovich-Szász variant based on Brenke type polynomials defined aswhere are strictly increasing sequences of positive numbers such that The classical Bernstein-Chlodowsky polynomials are defined bywhere and is a sequence of positive numbers with and In the last few decades the convergence estimation for linear positive operators is an active area of research amongst researchers. Several new operators have been introduced and their convergence behavior has been discussed (see [5–8]). In [9, 10] authors introduced a bivariate blending variant of the Szász type operators and studied local approximation properties for these operators. Also, they estimated the approximation order in terms of Peetre’s K-functional and partial moduli of continuity.

In the present paper, we define new bivariate operators associated with a combination of Szasz-Kantorovich-Chlodowsky operators based on Brenke polynomials as follows: where the sequences are defined as above and satisfy the following conditions: For operators defined in (36) we havewhere andIn this study, we give some basic convergence properties for the operators defined by (9) and study local approximation properties for these operators. Furthermore, we study the linear positive operators in a weighted space of function with two variables and estimate the rate of approximation of the operators in the terms of the weighted modulus of continuity.

#### 2. Notations and Auxiliary Results

We will subsequently need the following basic results to prove the main results.

In what follows, let , where is the two dimensional test functions.

By simple calculations we get the following lemma.

Lemma 1. *Let be the bivariate of Szasz-Kantorovich-Chlodowsky-Brenke operators defined by (9). For all , satisfy the following results:**(i)**(ii)**(iii)**(iv)**(v)**(vi)**(vii)*

*Proof. *In view of definition of operators defined by (9) we havewith the help of these equalities, we can easily prove required results.

Lemma 2. *It follows from Lemma 1 that*

*Proof. *The results follow from linearity of the operators and Lemma 1.

For sufficiently large , for all , by taking into consideration Lemma 1, and condition (10), we have the following equalities: andFurther, let , , and

#### 3. Main Results

To study the convergence of the sequence we shall use the following Korovkin type theorem, established by Volkov [11]. Next, the degree of approximation of the operator given by (36) will be established in the space of continuous function on compact set For , let , denote the space of all real valued continuous functions on , endowed with the norm

Theorem 3. *Let be the sequences of linear positive operators defined by (36). Then for each , we have** uniformly on the compact set *

*Proof. *From Lemma 1, we have anduniformly on The result follows from the well-known Volkov theorem.

*Example 4. *Let us consider the function For ; and ; ; the convergence of is illustrated in Figures 1(a) and 1(b), respectively. Further, in Table 1 we compute error estimation for operator (9) to the function