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# Statistical summability ( C , 1 ) and a Korovkin type approximation theorem

Syed A Mohiuddine1, Abdullah Alotaibi1* and Mohammad Mursaleen2

Author Affiliations

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

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

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Journal of Inequalities and Applications 2012, 2012:172 doi:10.1186/1029-242X-2012-172

 Received: 27 April 2012 Accepted: 19 July 2012 Published: 6 August 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

The concept of statistical summability has recently been introduced by Móricz [Jour. Math. Anal. Appl. 275, 277-287 (2002)]. In this paper, we use this notion of summability to prove the Korovkin type approximation theorem by using the test functions 1, , . We also give here the rate of statistical summability and apply the classical Baskakov operator to construct an example in support of our main result.

MSC: 41A10, 41A25, 41A36, 40A30, 40G15.

##### Keywords:
statistical convergence; statistical summability ; positive linear operator; Korovkin type approximation theorem

### 1 Introduction and preliminaries

In 1951, Fast [6] presented the following definition of statistical convergence for sequences of real numbers. Let , the set of all natural numbers and . Then the natural density of K is defined by if the limit exists, where the vertical bars indicate the number of elements in the enclosed set. The sequence is said to be statistically convergent to L if for every , the set has natural density zero, i.e., for each ,

In this case, we write . Note that every convergent sequence is statistically convergent but not conversely. Define the sequence by

(1.1)

Then x is statistically convergent to 0 but not convergent.

Recently, Móricz [12] has defined the concept of statistical summability as follows:

For a sequence , let us write . We say that a sequence is statistically summable if . In this case, we write .

In the following example, we exhibit that a sequence is statistically summable but not statistically convergent. Define the sequence by

(1.2)

Then

(1.3)

It is easy to see that and hence , i.e., a sequence is statistically summable to 0. On the other hand and , since the sequence is statistically convergent to 0. Hence is not statistically convergent.

Let be the space of all functions f continuous on . We know that is a Banach space with norm

The classical Korovkin approximation theorem is stated as follows [9]:

Let be a sequence of positive linear operators from into . Then , for all if and only if , for , where , and .

Recently, Mohiuddine [10] has obtained an application of almost convergence for single sequences in Korovkin-type approximation theorem and proved some related results. For the function of two variables, such type of approximation theorems are proved in [1] by using almost convergence of double sequences. Quite recently, in [13] and [14] the Korovkin type theorem is proved for statistical λ-convergence and statistical lacunary summability, respectively. For some recent work on this topic, we refer to [5,7,8,11,15,16]. Boyanov and Veselinov [3] have proved the Korovkin theorem on by using the test functions 1, , . In this paper, we generalize the result of Boyanov and Veselinov by using the notion of statistical summability and the same test functions 1, , . We also give an example to justify that our result is stronger than that of Boyanov and Veselinov [3].

### 2 Main result

Let be the Banach space with the uniform norm of all real-valued two dimensional continuous functions on ; provided that is finite. Suppose that . We write for ; and we say that L is a positive operator if for all .

The following statistical version of Boyanov and Veselinov’s result can be found in [4].

Theorem ALetbe a sequence of positive linear operators frominto. Then for all

if and only if

Now we prove the following result by using the notion of statistical summability .

Theorem 2.1Letbe a sequence of positive linear operators frominto. Then for all

(2.1)
if and only if

(2.2)

(2.3)

(2.4)

Proof Since each of 1, , belongs to , conditions (2.2)-(2.4) follow immediately from (2.1). Let . Then there exists a constant such that for . Therefore,

(2.5)

It is easy to prove that for a given there is a such that

(2.6)

whenever for all .

Using (2.5), (2.6), and putting , we get

This is,

Now, applying the operator to this inequality, since is monotone and linear, we obtain

Note that x is fixed and so is a constant number. Therefore,

(2.7)

Also,

(2.8)

It follows from (2.7) and (2.8) that

(2.9)

Now

Using (2.9), we obtain

Therefore,

since for all . Now taking , we get

(2.10)

where .

Now replacing by and then by in (2.10) on both sides. For a given choose such that . Define the following sets

Then , and so . Therefore, using conditions (2.2)-(2.4), we get

This completes the proof of the theorem. □

### 3 Rate of statistical summability

In this section, we study the rate of weighted statistical convergence of a sequence of positive linear operators defined from into .

Definition 3.1 Let be a positive nonincreasing sequence. We say that the sequence is statistically summable to the number L with the rate if for every ,

In this case, we write .

Now, we recall the notion of modulus of continuity. The modulus of continuity of , denoted by is defined by

It is well known that

(3.1)

Then we have the following result.

Theorem 3.2Letbe a sequence of positive linear operators frominto. Suppose that

(i) ,

(ii) , whereand.

Then for all, we have

where.

Proof Let and . From (2.8) and (3.1), we can write

Put . Hence, we get

where . Now replacing by

Using the Definition 3.1, and conditions (i) and (ii), we get the desired result. □

This completes the proof of the theorem.

### 4 Example and the concluding remark

In the following, we construct an example of a sequence of positive linear operators satisfying the conditions of Theorem 2.1 but does not satisfy the conditions of the Korovkin approximation theorem due to of Boyanov and Veselinov [3] and the conditions of Theorem A.

Consider the sequence of classical Baskakov operators [2]

where , .

Let be defined by

where the sequence is defined by (1.2). Note that this sequence is statistically summable to 0 but neither convergent nor statistically convergent. Now

we have that the sequence satisfies the conditions (2.2), (2.3), and (2.4). Hence, by Theorem 2.1, we have

On the other hand, we get , since , and hence

We see that does not satisfy the conditions of the theorem of Boyanov and Veselinov as well as of Theorem A, since is neither convergent nor statistically convergent.

Hence, our Theorem 2.1 is stronger than that of Boyanov and Veselinov [3] as well as Theorem A.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

### Acknowledgements

The authors would like to thank the Deanship of Scientific Research at King Abdulaziz University, Saudi Arabia, for its financial support under Grant No. 409/130/1432.

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