Journal of Function Spaces
Volume 2018 (2018), Article ID 6385451, 15 pages
https://doi.org/10.1155/2018/6385451

Research Article

On the Convergence of a Family of Chlodowsky Type Bernstein-Stancu-Schurer Operators

Lian-Ta Shu,¹ Guorong Zhou,² and Qing-Bo Cai¹

¹School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China
²School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China

Correspondence should be addressed to Qing-Bo Cai; qbcai@126.com

Received 26 April 2018; Revised 15 June 2018; Accepted 27 June 2018; Published 18 July 2018

Academic Editor: Gestur Ólafsson

Copyright © 2018 Lian-Ta Shu 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 construct a new family of univariate Chlodowsky type Bernstein-Stancu-Schurer operators and bivariate tensor product form. We obtain the estimates of moments and central moments of these operators, obtain weighted approximation theorem, establish local approximation theorems by the usual and the second order modulus of continuity, estimate the rate of convergence, give a convergence theorem for the Lipschitz continuous functions, and also obtain a Voronovskaja-type asymptotic formula. For the bivariate case, we give the rate of convergence by using the weighted modulus of continuity. We also give some graphs and numerical examples to illustrate the convergent properties of these operators to certain functions and show that the new ones have a better approximation to functions for one dimension.

1. Introduction

In 1912, Bernstein [1] proposed the famous polynomials called nowadays Bernstein polynomials to prove the Weierstrass approximation theorem. Later it was found that Bernstein polynomials possess many remarkable properties, which made them an area of intensive research. Even recently, there are also many papers mentioned about Bernstein type operators, such as [2–5]. A generalization of Bernstein polynomials based on -integers was proposed by Lupa in 1987 in [6]. However, the Lupas -Bernstein operators are rational functions rather than polynomials. The Phillips -Bernstein polynomials were introduced by Phillips in 1997 in [7]. In 2015, Mursaleen et al. [8] first introduced the (, )-analogue of Bernstein operators, in the case =1 these operators coincide with the Phillips -Bernstein operators. In the same year, they also proposed the (, )-analogue of Bernstein-Stancu operators in [9]. And then, in 2017, Khan et al. introduced the Lupa (, )-analogue of Bernstein operators. There are some recent papers relevant to Bernstein operators based on (, )-integers, such as [10–12]. Also some other positive operators related to (, )-integers; we listed some of them as [13–18].

In 1932, Chlodowsky introduced the classical Bernstein-Chlodowsky operators as

where and is a sequence of positive numbers such that , . These operators have been studied extensively, including one- and two-dimensional cases, which may be found in [19–25]. In 2017, Mishra et al. [26] introduced the Chlodowsky variant of (, ) Bernstein-Stancu-Schurer operators as

where , , with , , , and is an increasing sequence of positive terms with the properties and as . They discussed Korovkin-type approximation properties and rate of convergence of operators (2).

Due to the fact that these operators (2) reproduce only constant functions and it seems that there have been no two-dimensional case of their defined operators (2) at present, the first aim of this paper is to give a new type of these operators such that the new ones preserve not only constant functions but also linear functions; the second aim is to introduce the two-dimensional case based on these operators (3), which will be defined in (69). We also discuss weighted approximation properties of these new operators (3) and (69) and compare with the ones (2) by graphics and the absolute error bound of numerical analysis; we will show that the new ones (3) are better than (2) when approximating to functions .

We introduce new Chlodowsky type (, )-Bernstein-Stancu-Schurer operators as

and the basis function is defined as

where , , , and is an increasing sequence of positive terms with the properties , as .

We mention some definitions based on (, )-integers, and details can be found in [27–31]. For any fixed real number and , the (, )-integers are defined by

where denotes the -integers and . Also (, )-factorial and (, )-binomial coefficients are defined as follows:

The (, )-Binomial expansion is defined by

When , all the definitions of (, )-calculus above are reduced to -calculus.

The paper is organized as follows: In Section 2, we give some basic definitions regarding (, )-integers. In Section 3, we estimate the moments and central moments of these operators (3). In Section 4, we obtain weighted approximation theorem, establish local approximation theorems, estimate the rate of convergence, give a convergence theorem for the Lipschitz continuous functions, and also obtain a Voronovskaja-type asymptotic formula. In Section 5, we give some graphs and numerical examples to illustrate the convergent properties for one variable functions and compare with the ones in (2). In Section 6, we propose the bivariate case, give the rate of convergence by using the weighted modulus of continuity, and give some graphs and numerical analysis for two variables functions.

2. Auxiliary Results

Lemma 1 (see [26], Lemma 2). For , we have

Lemma 2. For , the following equalities hold:

Proof. From Lemma 1 and (3), we obtain (9), (10), and (11) by some simple computations, and we also get (12) and (13) by (9)–(11); here we omit it.

Remark 3. From (9) and (10) of Lemma 2, we know the operators preserve not only constant functions but also linear functions. That is to say, , where fixed .

Lemma 4 (see Theorem 2.1 of [32]). For , set , such that , , as . The following statements are true:

(A)If and , then .(B)If and , then .(C)If , and , then .

Lemma 5. Consider the sequences and , for satisfying the conditions (A), (B), or (C) of Lemma 4 and , ; then for fixed , the following equalities hold:

where and are functions depending on .

Proof. From (3), we have

and by some computations, we have

thus, using (4), (19), (20), and , we have

and, similarly, we have

Since , from (21), we get

due to

and, combining (18), (21)-(24), , (25), and (26), we obtain

Next, since

using the same methods and by some computations, we get

and, by formula (29), , and computations, we also have

Equation (16) can easily be obtained by (13). Finally, using the above conclusions and computations, we get

Lemma 5 is proved.

3. Approximation Properties

In the sequel, let , where and is an increasing sequence of positive terms with and . In order to obtain the weighted approximation Theorem 6, let , be sequences satisfying the conditions , , or of Lemma 4 and , .

Let be the set of all functions defined on satisfying the condition , where is the constant depending only on . We denote the subspace of all continuous functions belonging to by . Let be the subspace of all functions , for which is finite. The norm on is

Firstly, we discuss the weighted approximation theorem.

Theorem 6. For , we have

Proof. By using the Korovkin theorem, we see that it is sufficient to verify the following three conditions:

Since and , equality (34) holds true for and . Finally, for , from Lemma 2, we have

We can obtain by using Lemma 4. Theorem 6 is proved.

We give the following definitions: The space of all real valued continuous bounded functions defined on the interval is denoted by . The norm on is defined by . The Peetre’s -functional is given by

where and . For , the usual modulus of continuity and the second order modulus of smoothness are defined as follows:

By [33], there exists a constant , such that

Now, we establish local approximation theorems as follows.

Theorem 7. For , we have

where is a positive constant.

Proof. Let ; by Taylor’s expansion, we have

and, applying to (41), using (9) and (12), we get

Thus, from (13), we have

On the other hand, by (3) and (9), we have

Now (43) and (44) imply

and, from (36), taking infimum on the right hand side over all , we obtain

Finally, using (39), we get

Theorem 7 is proved.

Theorem 8. For and , we have

Proof. Since

applying to (49), we obtain

using Cauchy-Schwartz inequality, we have

Theorem 8 is proved.

Corollary 9. From Theorem 8, applied to , we have

Remark 10. For any fixed , we have , and this gives us a rate of pointwise convergence of the operators to .

Theorem 11. If is differentiable on and , then for , we have

Proof. Since and we have (12), we can write

and, by the mean value theorem, we obtain

so, by the Cauchy-Schwartz inequality, we get

and then we have the desired result by (13).

Corollary 12. From Theorem 11, let , applied to ; then we have

Next, we study the rate of convergence of the operators with the help of functions of Lipschitz class , where and . A function belongs to if

We have the following theorem.

Theorem 13. Let , ; we have

Proof. Obviously, are linear positive operators; since   , we have

Applying Hölder’s inequality for sums, we obtain

Theorem 13 is proved.

Now, we give a Voronovskaja-type asymptotic formula for .

Theorem 14. For , we have the following conclusion:

Proof. Let be fixed. By the Taylor formula, we may write

where is the Peano form of the remainder, ; using L’Hospital’s rule, we have

Since we have (12), applying to (63), we obtain

By the Cauchy-Schwarz inequality, we have

From and (17), we get . Hence, from (16), we have

Theorem 14 is proved.

Remark 15. For the function , from Theorem 14, we have the limit equality

which is the corresponding result in (16).

4. Graphical and Numerical Analysis I

In this section, we give several graphs and numerical examples to show the convergence of to with different values of parameters which satisfy the conclusions of Lemmas 4 and 5.

Example 16. Let ; the graphs of with , , , , , and different values of are shown in Figure 1. The graphs of with , , , , , and different values of are shown in Figure 2. The graphs of with , , , , and different values of and are shown in Figure 3. Moreover, we give a comparison on the approximation of (the red one) and (the yellow one) in Figure 4. In Tables 1 and 2, we show the absolute error bound of the approximation of with , , , and different values of , , and to .

Table 1: The absolute error bound of the approximation of with and different values of and .

0.215524
0.169329
0.132302
0.118517

0.202854
0.146061
0.101294
0.084213

0.201487
0.143719
0.098331
0.081025

0.201349
0.143485
0.098037
0.080710

Table 2: The absolute error bound of the approximation of with and different values of , , and , where .

0.20134927
0.14348509
0.09803662
0.08070974

0.20134028
0.14346631
0.09801166
0.08068261

0.20133938
0.14346444
0.09800917
0.08067989

0.20133930
0.14346425
0.09800892
0.08067962

0.20133929
0.14346423
0.09800889
0.08067960

Figure 1: The figures of with , , , , , and different values of .

Figure 2: The figures of with , , , , , and different values of .

Figure 3: The figures of with and different values of and .

Figure 4: The figures of (the red one) and (the yellow one) with .

5. Construction of Bivariate Operators and Weighted Approximation Properties

We introduce the bivariate tensor product (, )-analogue of Chlodowsky type Bernstein-Stancu-Schurer operators as follows:

where

, , and , are increasing sequences of positive terms with , as .

Lemma 17. Let , be the two-dimensional test functions; using Lemma 1, we easily obtain the following equalities:

Lemma 18. Using Lemma 17 and (16), (17) of Lemma 5, we have the following statements:

Let be the space of all functions defined on satisfying the condition , where is a positive constant depending only on and is a weighted function. We denote the subspace of all continuous functions belonging to by . Let be the subspace of all functions , for which is finite. The norm on is . For the infinite interval , , and , Ilarslan and Acar [34] introduced the weighted modulus of continuity as

which satisfies the following inequality:

From the definition of , we have

Now, we establish the degree approximation of operators in the weighted space by the weighted modulus of continuity .

Theorem 19. For , we have the following inequality:

where is a positive constant.

Proof. From (81) and (82), for , we get

and, applying the operators on the above inequality, we have

Using Cauchy-Schwarz inequality, we get

Using Lemma 18, we have

Then, we have

Let and ; we have

where are positive constants. Theorem 19 is proved.

6. Graphical and Numerical Analysis II

In this section, we give several graphs and numerical examples to show the convergence of to with different values of parameters which satisfy the conclusions of Lemmas 4 and 5.

Example 20. Let ; the graphs of with , and are shown in Figure 5. The graphs of with , and are shown in Figure 6. In Table 3, we show the absolute error bound of the approximation of with and different values of , where and , to .

Table 3: The absolute error bound of the approximation of with and different values of , where .

0.44324007
0.30755815
0.20568442
0.14177371

0.44321848
0.30751521
0.20562961
0.14171254

0.44321632
0.30751092
0.20562413
0.14170642

0.44321610
0.30751049
0.20562358
0.14170581

0.44321608
0.30751045
0.20562352
0.14170575

Figure 5: The figures of (the upper one) with and (the below one).

Figure 6: The figures of (the upper one) with and (the below one).

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11601266), the Natural Science Foundation of Fujian Province of China (Grant no. 2016J05017), and the Program for New Century Excellent Talents in Fujian Province University. The authors also thank Fujian Provincial Key Laboratory of Data Intensive Computing and Key Laboratory of Intelligent Computing and Information Processing of Fujian Province University.

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