Statistical summability and a Korovkin type approximation theorem
© Mohiuddine et al.; licensee Springer 2012
Received: 27 April 2012
Accepted: 19 July 2012
Published: 6 August 2012
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.
1 Introduction and preliminaries
Then x is statistically convergent to 0 but not convergent.
Recently, Móricz  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 .
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.
The classical Korovkin approximation theorem is stated as follows :
Let be a sequence of positive linear operators from into . Then , for all if and only if , for , where , and .
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 .
Theorem A Letbe a sequence of positive linear operators frominto. Then for all
Now we prove the following result by using the notion of statistical summability .
Theorem 2.1 Letbe a sequence of positive linear operators frominto. Then for all
whenever for all .
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 .
In this case, we write .
Then we have the following result.
, where and .
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.
where , .
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  as well as Theorem A.
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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