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

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.

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

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

Then

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].

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

if and only if

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,

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

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,

Also,

It follows from (2.7) and (2.8) that

Now

Using (2.9), we obtain

Therefore,

since for all . Now taking , we get

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. □

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

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.

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.

The authors declare that they have no competing interests.

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

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.

1. Anastassiou, GA, Mursaleen, M, Mohiuddine, SA: Some approximation theorems for functions of two variables through almost convergence of double sequences. J. Comput. Anal. Appl.. 13(1), 37–40 (2011)

2. Becker, M: Global approximation theorems for Szasz-Mirakjan and Baskakov operators in polynomial weight spaces. Indiana Univ. Math. J.. 27(1), 127–142 (1978). Publisher Full Text

3. Boyanov, BD, Veselinov, VM: A note on the approximation of functions in an infinite interval by linear positive operators. Bull. Math. Soc. Sci. Math. Roum.. 14(62), 9–13 (1970)

4. Duman, O, Demirci, K, Karakuş, S: Statistical approximation for infinite intervals. Preprint

5. Edely, OHH, Mohiuddine, SA, Noman, AK: Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett.. 23, 1382–1387 (2010). Publisher Full Text

6. Fast, H: Sur la convergence statistique. Colloq. Math.. 2, 241–244 (1951)

7. Gadjiev, AD, Orhan, C: Some approximation theorems via statistical convergence. Rocky Mt. J. Math.. 32, 129–138 (2002). Publisher Full Text

8. Karakuş, S, Demirci, K, Duman, O: Equi-statistical convergence of positive linear operators. J. Math. Anal. Appl.. 339, 1065–1072 (2008). Publisher Full Text

9. Korovkin, PP: Linear Operators and Approximation Theory, Hindustan Publ. Co., Delhi (1960)

10. Mohiuddine, SA: An application of almost convergence in approximation theorems. Appl. Math. Lett.. 24, 1856–1860 (2011). Publisher Full Text

11. Mohiuddine, SA, Alotaibi, A: Statistical convergence and approximation theorems for functions of two variables. J. Computational Analy. Appl. (accepted)

12. Móricz, F: Tauberian conditions under which statistical convergence follows from statistical summability . J. Math. Anal. Appl.. 275, 277–287 (2002). Publisher Full Text

13. Mursaleen, M, Alotaibi, A: Statistical summability and approximation by de la Vallée-Poussin mean. Appl. Math. Lett.. 24, 320–324 (2011). Publisher Full Text

14. Mursaleen, M, Alotaibi, A: Statistical lacunary summability and a Korovkin type approximation theorem. Ann. Univ. Ferrara. 57(2), 373–381 (2011). Publisher Full Text

15. Mursaleen, M, Karakaya, V, Ertürk, M, Gürsoy, F: Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput.. 218, 9132–9137 (2012). Publisher Full Text

16. Srivastava, HM, Mursaleen, M, Khan, A: Generalized equi-statistical convergence of positive linear operators and associated approximation theorems. Math. Comput. Model.. 55, 2040–2051 (2012). Publisher Full Text