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Advances in Mathematical Physics
Volume 2018, Article ID 3658389, 10 pages
https://doi.org/10.1155/2018/3658389
Research Article

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 [58]). 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

Table 1: Error estimation for operator (9) to the function for .
Figure 1

Example 5. For , the convergence of operators to function is illustrated in Figures 2(a) and 2(b), respectively, where , , and ; ; In Table 2 there are are compute error estimations for operator (9) to the function

Table 2: Error estimation for operator (9) to the function for .
Figure 2

An estimation of the rate of convergence can be obtained using the modulus of continuity for two dimensional real valued functions. Let and In what follows, we shall use the following modulus of continuity for bivariate real functions: Alternately, the complete modulus of continuity of which we denote by is defined as

Theorem 6. For any , then we have estimated where

Proof. From (9) and by definition of , we can write Using the Cauchy-Schwarz inequality, we obtainTaking , we obtain the desired result.
The partial modulus of continuity with respect to and is given by

Theorem 7. For any , then the inequalities satisfy where ,

Proof. Using the definition of partial modulus of continuity , we may write Consider . Using Lemma 1 and the well-known properties of the modulus of continuity, we have By using Cauchy-Schwarz inequality, we get So, by using (25), we obtainIn the same way we gain Hence from (39), (40), and (32), we arrive atFinally, choosing and , for all , we reach the desired result.
For , we define the Lipschitz class for bivariate case as follows: where , in , and is the Euclidean norm.

Theorem 8. Suppose that . Then, for every , we have where

Proof. First, we prove theorem for case Then, for and for each , using the monotonicity and linearity of operators, we may write where and
Using the Cauchy- Shwarz inequality and Lemma 2, the above inequality implies that Thus, the result holds for Secondly, let Then, for and for each , we get Now, applying Holder’s inequality with , , and Lemma 2, we get which leads us to the required result.

Theorem 9. If has continuous partial derivatives and and and denote the partial moduli of continuity of and respectively. Then we have estimate where are the positive constants such that

Proof. From the mean value theorem we have where and By using the above identity, we get Hence,Since and .
Using last inequalities, we have Now, applying the Cauchy-Schwarz inequality Now choosing and , we have This completes the proof.

4. Weighted Approximation Properties

The weighted Korovkin-type theorems are used for the purpose of this study, which are previously proved by Gadjiev [12, 13]. Therefore we need to introduce the notations of [13]. Let and be the space of all functions having the property , where and is a constant depending on function only. By we denote the subspace of all continuous functions belonging to It is clear that is a linear normed space with the norm Also, let be the subspace of all functions , for which , where

Theorem 10. Let belong to and Thenif and only if(i);(ii);(iii);(iv);(v); as for

Proof. The necessity part is trivial; then we need only to prove sufficiency. Let , and Since for each is uniformly on , for each there exists some , such that for each with implies Now let and and let be an arbitrary boundary point of such that Since is continuous on the boundary points also, then for each there exists such that implies On the other hand, if , we have