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  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 ,
Then x is statistically convergent to 0 but not convergent.
Recently, Móricz  has defined the concept of statistical summability as follows:
The classical Korovkin approximation theorem is stated as follows :
Recently, Mohiuddine  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  by using almost convergence of double sequences. Quite recently, in  and  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  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 .
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 .
It follows from (2.7) and (2.8) that
Using (2.9), we obtain
This completes the proof of the theorem. □
It is well known that
Then we have the following result.
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  and the conditions of Theorem A.
Consider the sequence of classical Baskakov operators 
Hence, our Theorem 2.1 is stronger than that of Boyanov and Veselinov  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.
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
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
Gadjiev, AD, Orhan, C: Some approximation theorems via statistical convergence. Rocky Mt. J. Math.. 32, 129–138 (2002). Publisher Full Text
Karakuş, S, Demirci, K, Duman, O: Equi-statistical convergence of positive linear operators. J. Math. Anal. Appl.. 339, 1065–1072 (2008). Publisher Full Text
Mohiuddine, SA: An application of almost convergence in approximation theorems. Appl. Math. Lett.. 24, 1856–1860 (2011). Publisher Full Text
Móricz, F: Tauberian conditions under which statistical convergence follows from statistical summability . J. Math. Anal. Appl.. 275, 277–287 (2002). Publisher Full Text
Mursaleen, M, Alotaibi, A: Statistical summability and approximation by de la Vallée-Poussin mean. Appl. Math. Lett.. 24, 320–324 (2011). Publisher Full Text
Mursaleen, M, Alotaibi, A: Statistical lacunary summability and a Korovkin type approximation theorem. Ann. Univ. Ferrara. 57(2), 373–381 (2011). Publisher Full Text
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
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