Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 719834, 7 pages

http://dx.doi.org/10.1155/2013/719834

## Generalized -Bernstein-Schurer Operators and Some Approximation Theorems

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 19 May 2013; Accepted 30 July 2013

Academic Editor: Simone Secchi

Copyright © 2013 M. Mursaleen and Asif Khan. 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 study statistical approximation properties of -Bernstein-Shurer operators and establish some direct theorems. Furthermore, we compute error estimation and show graphically the convergence for a function by operators and give its algorithm.

#### 1. Introduction and Preliminaries

In 1987, Lupaş [1] introduced the first -analogue of Bernstein operator and investigated its approximating and shape-preserving properties. Another -generalization of the classical Bernstein polynomials is due to Phillips [2]. After that many generalizations of well-known positive linear operators, based on -integers, were introduced and studied by several authors. Recently the statistical approximation properties have also been investigated for -analogue polynomials. For instance, in [1] -analogues of Bernstein-Kantorovich operators; in [3] -Baskakov-Kantorovich operators; in [4] -Szász-Mirakjan operators; in [5, 6] -Bleimann, Butzer and Hahn operators; in [7] -analogue of Baskakov and Baskakov-Kantorovich operators; in [8] -analogue of Szász Kantorovich operators; in [9, 10] -analogue of Stancu-Beta operators; and in [11] -Lagrange polynomials were defined and their classical approximation or statistical approximation properties were investigated.

Schurer [12] introduced the following operators defined for any and any function :

Recently, Muraru [13] introduced the -analogue of these operators and investigated their approximation properties and rate of convergence using modulus of continuity. Note that Radu [14] has also used -intgers to define and study the approximation properties of the -analogue of Kantorovich operators.

In this paper, we study the statistical approximation properties by -Bernstein-Schurer operators. We also give some direct theorems.

We recall certain notations of -calculus. Let . For any , the -integer is defined by and the -factorial by Also, the -binomial coefficients are defined by Details on -integers can be found in [14].

Muraru [13] introduced the following operators known as the generalized -Bernstein-Schurer operators. For any , a fixed positive integer, and ,

We note the following properties as in [13] for .

Lemma 1. *For , ,
*

Lemma 2. *For , ,
*

Lemma 3. *For , ,
*

Lemma 4. *Let . Then for *

Lemma 5. *Let . Then for *

#### 2. Statistical Approximation

In this section we obtain the Korovkin type weighted statistical approximation properties for these operators defined in (5). Korovkin type approximation theory [15] has also many useful connections, other than classical approximation theory, in other branches of mathematics (see Altomare and Campiti [16]).

First we recall the concept of statistical convergence for sequences of real numbers which was introduced by Fast [17] and further studied by many others.

Let and . Then the *natural density* of is defined by if the limit exists, where denotes the cardinality of the set .

A sequence of real numbers is said to be to provided that for every the set has natural density zero; that is, for each ,

In this case, we write . Note that every convergent sequence is statistically convergent but not conversely, even unbounded sequence may be statistically convergent. For example, let be defined by

Then −, but is not convergent.

Recently the idea of statistical convergence has been used in proving some approximation theorems, in particular, Korovkin type approximation theorems by various authors, and it was found that the statistical versions are stronger than the classical ones. Authors have used many types of classical operators and test functions to study the Korovkin type approximation theorems which further motivate continuation of this study. After the paper of Gadjiev and Orhan [18], different types of summability methods have been deployed in approximation process, for example, [19–23]. Recently, -statistical convergence has been used to the summability of Walsh-Fourier series [24].

Let be the space of all bounded and continuous functions on . Then is a normed linear space with . Let be a function of the type of modulus of continuity. The principal properties of the function are the following: is a nonnegative increasing function on ,.

Let be the space of all real valued functions defined on satisfying the following condition: for any .

We consider a sequence , , such that The condition (14) guarantees that as .

Now our first result is as follows.

Theorem 6. *Let be the sequence of the operators (5), and the sequence satisfies (14). Then for any function ,
*

*Proof. *Let where . Since , therefore we can write
as
By (14), it can be observed that
Similarly
For a given , let us define the following sets:
It is obvious that ; it can be written as
By using (14), we get
Therefore
and we have
Lastly, we have
Therefore by (25), we get
Now, if we choose
then, by (14), we can write
Now for given , we define the following four sets:
It is obvious that . Then we obtain
Using (28), we get
Since
we get
which implies that

This completes the proof of the theorem.

*Remark 7. *In the following example, we demonstrate that the statistical version is stronger than the ordinary approximation. Let us write , where the sequence is defined by (12). Then under the hypothesis of the previous theorem, we have
However, does not exist, since is statistically convergent but not convergent.

#### 3. Direct Theorems

The Peetre’s -functional is defined by where

By [25], there exists a positive constant such that , , where the second-order modulus of continuity is given by

Also for the usual modulus of continuity is given by

Theorem 8. *Let and such that . Then for all and fixed, there exists an absolute constant such that
**
where
*

*Proof. *Let . From Taylor’s expansion, we get
And, by Lemmas 1, 2, and 3, we get
Using Lemma 5, we obtain
On the other hand, by the definition of , we have
Now
Hence taking infimum on the right hand side over all , we get
In view of the property of -functional, we get

This completes the proof of the theorem.

Let be the space of all bounded functions for which is finite.

Theorem 9. *Let be such that , , and the sequence satisfies (14). Then the following equality holds:
**
uniformly on .*

*Proof. *By the Taylor’s formula we may write
where is the remainder term and . Applying to (50), we obtain
By the Cauchy-Schwartz inequality, we have
Observing that and , then it follows from Theorem 8 that
uniformly with respect to . Now from (52), (53), and Lemma 5, we get
Finally using Lemmas 4 and 5, we get the following:

This completes the proof of the theorem.

#### 4. Example

Let us take . We compute error estimation [26] by using modulus of continuity for operators (5) to the function shown in Table 1 with the help of MATLAB and Algorithm 1.

For , 30, and 50, the convergence of operators (5) to function is illustrated in Figure 1 with help of Algorithm 2.

#### References

- A. Lupaş, “A
*q*-analogue of the Bernstein operator,”*Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca*, vol. 9, pp. 85–92, 1987. View at Google Scholar - G. M. Phillips, “Bernstein polynomials based on the q-integers, The heritage of P.L. Chebyshev,”
*Annals of Numerical Mathematics*, vol. 4, pp. 511–518, 1997. View at Google Scholar - V. Gupta and C. Radu, “Statistical approximation properties of q-Baskakov-Kantorovich operators,”
*Central European Journal of Mathematics*, vol. 7, no. 4, pp. 809–818, 2009. View at Publisher · View at Google Scholar · View at Scopus - M. Örkcü and O. Doǧru, “Weighted statistical approximation by kantorovich type
*q*-Szász-Mirakjan operators,”*Applied Mathematics and Computation*, vol. 217, no. 20, pp. 7913–7919, 2011. View at Publisher · View at Google Scholar · View at Scopus - A. Aral and O. Dğru, “Bleimann, Butzer, and Hahn operators based on the q-integers,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 79410, 12 pages, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. Ersan and O. Doğru, “Statistical approximation properties of q-Bleimann, Butzer and Hahn operators,”
*Mathematical and Computer Modelling*, vol. 49, no. 7-8, pp. 1595–1606, 2009. View at Publisher · View at Google Scholar · View at Scopus - N. I. Mahmudov, “Statistical approximation of Baskakov and Baskakov-Kantorovich operators based on the $q$-integers,”
*Central European Journal of Mathematics*, vol. 8, no. 4, pp. 816–826, 2010. View at Publisher · View at Google Scholar · View at Scopus - N. Mahmudov and V. Gupta, “On certain
*q*-analogue of Szász Kantorovich operators,”*Journal of Applied Mathematics and Computing*, vol. 37, no. 1-2, pp. 407–419, 2011. View at Publisher · View at Google Scholar · View at Scopus - A. Aral and V. Gupta, “On the
*q*-analogue of Stancu-Beta operators,”*Applied Mathematics Letters*, vol. 25, no. 1, pp. 67–71, 2012. View at Publisher · View at Google Scholar · View at Scopus - M. Mursaleen and A. Khan, “Statistical approximation properties of modified
*q*-Stancu-Beta operators,”*Bulletin of the Malaysian Mathematical Sciences Society*, vol. 36, no. 3, pp. 683–690, 2013. View at Google Scholar - M. Mursaleen, A. Khan, H. M. Srivastava, and K. S. Nisar, “Operators constructed by means of q-Lagrange polynomials and A-statistical approximation,”
*Applied Mathematics and Computation*, vol. 219, pp. 6911–6918, 2013. View at Google Scholar - F. Schurer, “Linear positive operators in approximation theory,” Tech. Rep., Mathematical Institute Delft University of Technology, 1962. View at Google Scholar
- C.-V. Muraru, “Note on
*q*-Bernstein-Schurer operators,”*Studia Universitatis Babeş-Bolyai, Mathematica*, vol. 56, no. 2, pp. 489–495, 2011. View at Google Scholar - C. Radu, “Statistical approximation properties of Kantorovich operators based on
*q*-integers,”*Creative Mathematics and Informatics*, vol. 17, no. 2, pp. 75–84, 2008. View at Google Scholar - P. P. Korovkin,
*Linear Operators and Approximation Theory*, Hindustan Publishing Corporation, Delhi, India, 1960. - F. Altomare and M. Campiti,
*Korovkin Type Approximation Theory and Its Applications*, vol. 17 of*Studia Mathematica*, Walter de Gruyter, Berlin, Germany, 1994. - H. Fast, “Sur la convergence statistique,”
*Colloquium Mathematicum*, vol. 2, pp. 241–244, 1951. View at Google Scholar - A. D. Gadjiev and C. Orhan, “Some approximation theorems via statistical convergence,”
*Rocky Mountain Journal of Mathematics*, vol. 32, no. 1, pp. 129–138, 2002. View at Google Scholar · View at Scopus - G. A. Anastassiou, M. Mursaleen, and S. A. Mohiuddine, “Some approximation theorems for functions of two variables through almost convergence of double sequences,”
*Journal of Computational Analysis and Applications*, vol. 13, no. 1, pp. 37–46, 2011. View at Google Scholar · View at Scopus - M. Mursaleen and A. Alotaibi, “Statistical summability and approximation by de la Vallée-Poussin mean,”
*Applied Mathematics Letters*, vol. 24, pp. 320–324, 2012, Erratum: Applied Mathematics Letters, vol. 25, pp. 665, 2012. View at Publisher · View at Google Scholar · View at Scopus - S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “Statistical summability (C, 1) and a Korovkin type approximation theorem,”
*Journal of Inequalities and Applications*, vol. 2012, article 172, 2012. View at Google Scholar - M. Mursaleen, V. Karakaya, M. Ertürk, and F. Gürsoy, “Weighted statistical convergence and its application to Korovkin type approximation theorem,”
*Applied Mathematics and Computation*, vol. 218, no. 18, pp. 9132–9137, 2012. View at Publisher · View at Google Scholar · View at Scopus - M. Mursaleen and A. Kilicman, “Korovkin second theorem via
*B*-statistical*A*-summability,”*Abstract and Applied Analysis*, vol. 2013, Article ID 598963, 6 pages, 2013. View at Publisher · View at Google Scholar - A. Alotaibi and M. Mursaleen, “A-statistical summability of Fourier series and Walsh-Fourier series,”
*Applied Mathematics & Information Sciences*, vol. 6, no. 3, pp. 535–538, 2012. View at Google Scholar - R. A. Devore and G. G. Lorentz,
*Constructive Approximation*, Springer, Berlin, Germany, 1993. - S. Sezgin and Y. Ertan, “Rate of convergence for Szász type operators including Sheffer polynomials,”
*Studia Universitatis Babeş-Bolyai, Mathematica*, vol. 58, pp. 55–63, 2013. View at Google Scholar