Korovkin second theorem via statistical summability
© Mohiuddine and Alotaibi; licensee Springer. 2013
Received: 25 July 2012
Accepted: 18 March 2013
Published: 3 April 2013
Korovkin type approximation theorems are useful tools to check whether a given sequence of positive linear operators on of all continuous functions on the real interval is an approximation process. That is, these theorems exhibit a variety of test functions, which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, x and in the space as well as for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line. In this paper, we use the notion of statistical summability to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence.
MSC:41A10, 41A25, 41A36, 40A30, 40G15.
Keywordsdensity statistical convergence statistical summability positive linear operator Korovkin type approximation theorem
1 Introduction and preliminaries
In this case, we write . Note that every convergent sequence is statistically convergent but not conversely.
Recently, Móricz  has defined the concept of statistical summability as follows:
For a sequence , let us write . We say that a sequence is statistiallly summable if . In this case we write .
Note that evey -summable sequence is also statistiallly summable to the same limit but not conversely.
Then this sequence is neither convergent nor statistically convergent. But x is -summable to 0, and hence statistiallly summable to 0.
Theorem I Let be a sequence of positive linear operators from into . Then , for all if and only if , for , where , and .
Theorem II Let be a sequence of positive linear operators from into . Then , for all if and only if , for , where , and .
Several mathematicians have worked on extending or generalizing the Korovkin’s theorems in many ways and to several settings, including function spaces, abstract Banach lattices, Banach algebras, Banach spaces and so on. This theory is very useful in real analysis, functional analysis, harmonic analysis, measure theory, probability theory, summability theory and partial differential equations. But the foremost applications are concerned with constructive approximation theory, which uses it as a valuable tool. Even today, the development of Korovkin-type approximation theory is far from complete. Note that the first and the second theorems of Korovkin are actually equivalent to the algebraic and the trigonometric version, respectively, of the classical Weierstrass approximation theorem . Recently, the Korovkin second theorem is proved in  by using the concept of statistical convergence. Quite recently, Korovkin second theorem is proved by Demirci and Dirik  for statistical σ-convergence . For some recent work on this topic, we refer to [24–33]. In this work, we prove Korovkin second theorem by applying the notion of statistical summability . We also give an example to justify that our result is stronger than Theorem II.
2 Main result
We write for ; and we say that L is a positive operator if for all .
This completes the proof of the theorem. □
3 Rate of weighted statistical convergence
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 .
As usual we have the following auxiliary result whose proof is standard.
- (i), for any scalar α,
Then we have the following result.
- (ii), where and .
Now, using 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 Theorem II.
The sequence is a positive kernel which is called the Fejér kernel, and the corresponding operators , , are called the Fejér convolution operators.
Note that the Theorem II is satisfied for the sequence . In fact, we have
i.e. our theorem holds. But on the other hand, Theorem II does not hold for our operator defined by (4.1), since the sequence is not convergent.
Hence, our Theorem 2.1 is stronger than that of Theorem II.
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (410/130/1432). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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