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Journal of Function Spaces
Volume 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

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; moc.621@iacbq

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 [25]. 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 [1012]. Also some other positive operators related to (, )-integers; we listed some of them as [1318].

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 [1925]. In 2017, Mishra et al. [26] introduced the Chlodowsky variant of (, ) Bernstein-Stancu-Schurer operators aswhere , , 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 asand the basis function is defined aswhere , , , 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 [2731]. 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 haveand by some computations, we havethus, using (4), (19), (20), and , we haveand, similarly, we haveSince , from (21), we getdue toand, combining (18), (21)-(24), , (25), and (26), we obtainNext, since using the same methods and by some computations, we getand, by formula (29), , and computations, we also haveEquation (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 haveWe 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 bywhere 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 havewhere is a positive constant.

Proof. Let ; by Taylor’s expansion, we haveand, applying to (41), using (9) and (12), we getThus, from (13), we haveOn the other hand, by (3) and (9), we haveNow (43) and (44) implyand, from (36), taking infimum on the right hand side over all , we obtainFinally, using (39), we getTheorem 7 is proved.

Theorem 8. For and , we have

Proof. Sinceapplying to (49), we obtainusing Cauchy-Schwartz inequality, we haveTheorem 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 writeand, by the mean value theorem, we obtain so, by the Cauchy-Schwartz inequality, we getand 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 obtainTheorem 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 writewhere is the Peano form of the remainder, ; using L’Hospital’s rule, we haveSince we have (12), applying to (63), we obtain