Journal of Function Spaces and Applications

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

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

Research Article

Generalized -Bernstein-Schurer Operators and Some Approximation Theorems

M. Mursaleen and Asif Khan

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.

Table 1: Error estimation table.

(for , )
Error bound at
Error bound at
Error bound at

20
0.2309
1.5709
2.7170

30
0.1893
1.1865
1.5829

50
0.1750
1.0835
1.3792

  ;

 [];

;   0.9;

;

    1 : 3

      ;

                        ;

     ;

     ;

     ;

     ;

end

Algorithm 1: Error estimate algorithm.

For , 30, and 50, the convergence of operators (5) to function

is illustrated in Figure 1 with help of Algorithm 2.

 [];

   1 : 3

     ;

       [1 : 100];

     ;

        0 : 0.01 : 1

         ;

            0 :  m 

             ;

                0 :  m 

                 ;

                    ;

          ;

          ;

       0 : 0.01 : 1;

          “Color”, “blue”, “LineWidth”,

          g”

          “Color” “green” “LineWidth”,

          r”

          “Color” “red” “LineWidth”

  [0 : 0.01 : 1];

;

k”

Algorithm 2: For the generalized -Bernstein-Schurer operators.

Figure 1

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