Approximation by One and Two Variables of the Bernstein-Schurer-Type Operators and Associated GBS Operators on Symmetrical Mobile Interval

Aslan, Reşat; İzgi, Aydın

doi:https://doi.org/10.1155/2021/9979286

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Volume 2021 | Article ID 9979286 | https://doi.org/10.1155/2021/9979286

Approximation by One and Two Variables of the Bernstein-Schurer-Type Operators and Associated GBS Operators on Symmetrical Mobile Interval

Reşat Aslan¹and Aydın İzgi¹

Academic Editor: Tuncer Acar

Received05 Mar 2021

Revised31 Mar 2021

Accepted10 Apr 2021

Published04 May 2021

Abstract

In this article, we purpose to study some approximation properties of the one and two variables of the Bernstein-Schurer-type operators and associated GBS (Generalized Boolean Sum) operators on a symmetrical mobile interval. Firstly, we define the univariate Bernstein-Schurer-type operators and obtain some preliminary results such as moments, central moments, in connection with a modulus of continuity, the degree of convergence, and Korovkin-type approximation theorem. Also, we derive the Voronovskaya-type asymptotic theorem. Further, we construct the bivariate of this newly defined operator, discuss the order of convergence with regard to Peetre’s -functional, and obtain the Voronovskaya-type asymptotic theorem. In addition, we consider the associated GBS-type operators and estimate the order of approximation with the aid of mixed modulus of smoothness. Finally, with the help of the Maple software, we present the comparisons of the convergence of the bivariate Bernstein-Schurer-type and associated GBS operators to certain functions with some graphical illustrations and error estimation tables.

1. Introduction

In [1], Bernstein suggested his polynomials that still inspire many studies today as follows: for any and any .

In 1962, the operators , which are called Bernstein-Schurer, are proposed by Schurer [2] as follows: for any and .

Very recently, many modifications and generalizations of the Bernstein or Bernstein-Schurer operators for univariate and bivariate cases are discussed by many authors. For instance, Acar et al. [3] established local and global approximation results in terms of modulus of continuity for a new type of the Bernstein-Durrmeyer operators on mobile interval. Izgi [4] presented a new type of the Bernstein polynomials and studied several approximation results of the univariate and bivariate of these operators. For the parameter , Chen et al. [5] defined a new generalization of the Bernstein operator and derived the order of convergence and Voronovskaya-type asymptotic relation for the Bernstein operator. Kajla and Acar [6] constructed a new kind of the Bernstein operator and studied a uniform convergence estimate, some direct results involving the asymptotic theorems for these operators. Acar et al. [7] introduced the Kantorovich modifications of the Bernstein operators for bivariate functions using a new integral and obtained the uniform convergence and rate of approximation in terms of modulus of continuity for these operators. Further, for , Cai [8] introduced the Bézier version of the Kantorovich-type Bernstein polynomials and gained the global and direct approximation theorems. Acar and Kajla [9] introduced an extension of the bivariate generalized Bernstein operators with nonnegative real parameters and studied the degree of approximation with regard to Peetre’s -functional and Lipschitz-type functions. Bărbosu [10] demonstrated the uniform convergence and estimated the degree of approximation of the bivariate of the Bernstein-Schurer operators. Căbulea [11] considered the generalizations of the Kantorovich and Durrmeyer type of the Bernstein-Schurer operators and evaluated in connection with the modulus of continuity the order of approximation of these operators. Also, for some recent works, we can refer the readers to ([12–23]).

By the motivation of the all the above-mentioned works, we define the univariate Bernstein-Schurer-type operators on a symmetrical mobile interval. Let the intervals be , , and be the set of all continuous and bounded functions on For a function and the univariate Bernstein-Schurer operators are defined as where .

The goal of the present work is to obtain some approximation features of the operators given by (3). We show the uniform convergence, estimate the degree of convergence with the help of modulus of continuity, and prove the Voronovskaya-type asymptotic theorem for the (3) operators. Next, we define the bivariate of (3) operators, compute the order of convergence by using Peetre’s -functional, and derive the Voronovskaya-type asymptotic theorem for the bivariate case. Further, we construct the associated GBS type of bivariate operators and estimate their degree of convergence in terms of mixed modulus of smoothness. Finally, by the help of the Maple software, we give comparisons of the convergence of bivariate of (3) operators and related GBS operators to the certain functions with some graphics and error estimation tables.

2. Main Results

Lemma 1. Let the operators be defined by (3). Then, for all , the following moments verify

Proof. From (3), it becomes The last two identities can be obtained by applying similar methods; hence, we have omitted the details.

Lemma 2. For all , we obtain the following central moments:

Proof. The proof of this lemma can be directly obtained by using the linearity of (3) operators and as a consequence of Lemma 1.

Corollary 3. For all , the following identities hold: In the next theorem we show the uniform convergence of (3) operators. As it is known, the space denotes the real-valued continuous functions on , and it is equipped with the norm for a function as follows:

Theorem 4. Let the operators be given by (3). Then, for all , converges to uniformly on .

Proof. From (4), it is obvious that By (5), we arrive Similarly, using (6), then Hence, according to the Korovkin theorem [24], the (3) operators converge uniformly to on

Further, we will obtain the degree of approximation of (3) operators. Let the modulus of continuity for a function be given by

Since has some useful properties which can be found in [25].

Theorem 5. Let Then, for every , the following inequality is verified: where

Proof. Taking into account the following common property of the modulus of continuity: by the linearity of the operator (3), then Utilizing the Cauchy-Schwarz inequality yields If we take thus Theorem 5 is proven.

Theorem 6. Let the operators be given by (3). Then, for any such that , the following identity holds: uniformly on

Proof. Suppose that and are fixed. By the Taylor formula, hence where is a form of Peano of the rest term, and since
Operating to (22), then From Lemma 2, it becomes Applying the Cauchy-Schwarz inequality, one has Owing to thus uniformly on with Theorem 4.
Combining (25) and (26) and by Lemma 2, we get Hence, which gives the proof.

3. Construction of the Bivariate Bernstein-Schurer-Type Operators

Let the intervals be , and by , we denote the set of all real-valued continuous functions on , and it is equipped with the norm

We define the bivariate of operators given by (3) as follows: where , and

It can be seen that the operators given by (29) are positive and linear.

Lemma 7. Let with , be the bivariate test functions. Then, one has

Proof. The proof of the above equalities can be reached easily as a consequence of Lemma 1 and by (29); hence, we have omitted the details.

Corollary 8. In view of Lemma 7, the following relations hold true:

Theorem 9. Let the operators be given by (29). Then for any , we arrive at

Proof. It is seen from the following that as Thus, these results complete the proof, as required by the Volkov theorem [26].

Moreover, for the operators given by (29), we want to derive the Voronovskaya-type asymptotic theorem and estimate the degree of convergence with the help of Peetre’s -functional.

Suppose that represents the space of all functions of such that . The norm on and Peetre’s -functional of are given as follows, respectively. where .

For a constant the following inequality holds, where denotes the second order of the modulus of continuity of Also, for the ordinary modulus of continuity is defined as

Theorem 10. Suppose that . Then, the following relation holds uniformly on

Proof. For the arbitrary using the Taylor formula, it becomes for where and as
Operating on (38) yields If we use the Cauchy-Schwarz inequality to the last part of (39), one has Considering Theorem 9 and because of as then uniformly on Also, from Corollary 8, it is clear to see Further, by Corollary 3, it follows that Thus, Hence, the desired sequel is arrived as follows: uniformly on .

Theorem 11. Suppose that . Then, the following inequality is verified: where is a constant and independent of and and , .

Proof. Let us define the following auxiliary operators: By Lemma 7, we obtain Suppose that and ; by the Taylor formula, we may write Now, operating on (49), we get Hence, If we choose then Also, by Lemma 7 and (52), we arrive at Next, from (53) Consequently, on (54), utilizing the infimum on the right-hand side over all and by (35), it becomes which ends the proof.

4. Construction of the GBS Type of

The notion of the -continuous and -differentiable functions was firstly used by Bögel [27, 28]. Dobrescu and Matei [29] considered the GBS (Generalized Boolean Sum) kind of the bivariate of the Bernstein polynomials. Next, using the -continuous functions by the GBS operators, which is related to quantitative version of the Korovkin-type convergence theorem, firstly has been improved by Badea et al. [30, 31]. Pop and Fărcas [32] obtained some approximation of the -continuous and -differentiable functions by GBS type of the Bernstein bivariate operators. İspir [33] established quantitative estimates for the GBS of the Chlodowsky-Szász-kind operators. Recently, some authors introduced the GBS operators of various operators (we refer the readers to [34–42]).

Let a function where , are compact real intervals of With the mixed difference of the function is given as

A function is named Bögel-continuous (-continuous) at if

A function is named Bögel-differentiable (-differentiable) at if the below result which denoted by exists and finite

Note that, by and , we represent the sets of all -differentiable and -continuous functions on respectively.

A function is named Bögel-bounded (bounded) on if there consists such that for every

Also, if is a compact subset of , hence all -continuous functions are bounded on

Further, by , we represent the set of all -bounded functions on and equipped with the norm Considering the definition of -continuous, then (see details by [43]).

The mixed modulus of smoothness for is given by where , and . Also, for all , the following inequality satisfy

More details about the mixed modulus of smoothness can be found in [30, 31].

Now, for all , for any and , , we construct the GBS-type operators given by (29) operators as follows:

Exactly, for any and for all , the GBS type of operators is given as where is defined as in (29).

It is clear that the operators given by (62) are linear and positive. In the following theorem, with regard to the mixed modulus of smoothness, we estimate the order of approximation of the (62) operators.

Theorem 12. For all and for each the operators given by (62) satisfy the following inequality: where and

Proof. In view of (60), it gives for all , and for any By (56), we have Operating and by the definition of , then It follows that From (64) and by utilizing the Cauchy-Schwarz inequality, one has Taking into account Corollary 8, it becomes Hence, If we choose and , then the desired result is arrived.

5. Graphics and Error Estimation Tables

In this section, with the help of the Maple software, we give some plots and error estimation tables for the comparison of the convergence behavior of (29) and (62) operators to the certain functions.

Example 1. Let . In Figure 1, for (red), (green), (blue), and by choosing , we illustrate the convergence of (29) operators to (yellow). Also, in Table 1, we determine the error of approximation (29) operators to for the certain values of and , respectively. It is clear from Table 1 that as the values of and increase then the error of approximation (29) operators to decreases.

Example 2. Let . In Figure 2, for , , we compare the convergence of (29) (red) and the associated GBS operators (62) (blue) to (yellow). Also, in Table 2, we evaluate the error of approximation (29) and (62) operators to for , , and the certain points of . It is obvious that, for , , the absolute difference between (29) operators to is greater than that of (62) operators to . Namely, the (62) operators has better approximation than (29) operators.

Data Availability

All data required for this paper are included within this paper.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All the authors contributed equally and significantly in writing this article.

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Copyright © 2021 Reşat Aslan and Aydın İzgi. 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.

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