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.
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.
- Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241–244.MathSciNetGoogle Scholar
- Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 1951, 2: 73–74.MathSciNetGoogle Scholar
- Móricz F:Tauberian conditions under whnih statistical corvergence follows fron statistical summability. J. Math. Anal. Appl. 2002, 275: 277–287. 10.1016/S0022-247X(02)00338-4MathSciNetView ArticleGoogle Scholar
- Mohiuddine SA, Aiyub M: Lacunary statistical convergence in random 2-normed spaces. Appl. Math. Inform. Sci. 2012, 6(3):581–585.MathSciNetGoogle Scholar
- Mohiuddine SA, Alghamdi MA: Statistical summability through a lacunary sequence in locally solid Riesz spaces. J. Inequal. Appl. 2012, 2012: 225. 10.1186/1029-242X-2012-225MathSciNetView ArticleGoogle Scholar
- Mohuiddine SA, Alotaibi A, Alsulami SM: Ideal convergence of double sequences in random 2-normed spaces. Adv. Differ. Equ. 2012, 2012: 149. 10.1186/1687-1847-2012-149View ArticleGoogle Scholar
- Mohiuddine SA, Alotaibi A, Mursaleen M: Statistical convergence of double sequences in locally solid Riesz spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 719729Google Scholar
- Mohiuddine SA, Lohani QMD: On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos Solitons Fractals 2009, 42: 1731–1737. 10.1016/j.chaos.2009.03.086MathSciNetView ArticleGoogle Scholar
- Mohiuddine SA, Savaş E: Lacunary statistically convergent double sequences in probabilistic normed spaces. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 2012, 58: 331–339. 10.1007/s11565-012-0157-5View ArticleGoogle Scholar
- Mohiuddine SA, Şevli H, Cancan M: Statistical convergence in fuzzy 2-normed space. J. Comput. Anal. Appl. 2010, 12(4):787–798.MathSciNetGoogle Scholar
- Mursaleen M, Mohiuddine SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 41: 2414–2421. 10.1016/j.chaos.2008.09.018MathSciNetView ArticleGoogle Scholar
- Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2009, 233: 142–149. 10.1016/j.cam.2009.07.005MathSciNetView ArticleGoogle Scholar
- Mursaleen M, Mohiuddine SA: On ideal convergence of double sequences in probabilistic normed spaces. Mathem. Rep. (Buchar.) 2010, 12(62)(4):359–371.MathSciNetGoogle Scholar
- Mursaleen M, Mohiuddine SA: On ideal convergence in probabilistic normed spaces. Math. Slovaca 2012, 62: 49–62. 10.2478/s12175-011-0071-9MathSciNetView ArticleGoogle Scholar
- Mursaleen M, Mohiuddine SA, Edely OHH: On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces. Comput. Math. Appl. 2010, 59: 603–611. 10.1016/j.camwa.2009.11.002MathSciNetView ArticleGoogle Scholar
- Savaş E, Mohiuddine SA:-statistically convergent double sequences in probabilistic normed spaces. Math. Slovaca 2012, 62(1):99–108. 10.2478/s12175-011-0075-5MathSciNetGoogle Scholar
- Mursaleen M, Çakan C, Mohiuddine SA, Savaş E: Generalized statistical convergence and statistical core of double sequences. Acta Math. Sin. Engl. Ser. 2010, 26: 2131–2144. 10.1007/s10114-010-9050-2MathSciNetView ArticleGoogle Scholar
- Korovkin PP: Convergence of linear positive operators in the spaces of continuous functions. Dokl. Akad. Nauk SSSR 1953, 90: 961–964. (in Russian)MathSciNetGoogle Scholar
- Korovkin PP: Linear Operators and Approximation Theory. Hindustan Publ. Co., Delhi; 1960.Google Scholar
- Altomare F: Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory 2010, 5: 92–164.MathSciNetGoogle Scholar
- Duman O: Statistical approximation for periodic functions. Demonstr. Math. 2003, 36: 873–878.MathSciNetGoogle Scholar
- Demirci K, Dirik F: Approximation for periodic functions via statistical σ -convergence. Math. Commun. 2011, 16: 77–84.MathSciNetGoogle Scholar
- Mursaleen M, Edely OHH: On invariant mean and statistical convergence. Appl. Math. Lett. 2009, 22: 1700–1704. 10.1016/j.aml.2009.06.005MathSciNetView ArticleGoogle Scholar
- Anastassiou GA, Mursaleen M, Mohiuddine SA: Some approximation theorems for functions of two variables through almost convergence of double sequences. J. Comput. Anal. Appl. 2011, 13(1):37–46.MathSciNetGoogle Scholar
- Belen C, Mohiuddine SA: Generalized weighted statistical convergence and application. Appl. Math. Comput. 2013. doi:10.1016/j.amc.2013.03.115Google Scholar
- Edely OHH, Mohiuddine SA, Noman AK: Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett. 2010, 23: 1382–1387. 10.1016/j.aml.2010.07.004MathSciNetView ArticleGoogle Scholar
- Mohiuddine SA: An application of almost convergence in approximation theorems. Appl. Math. Lett. 2011, 24: 1856–1860. 10.1016/j.aml.2011.05.006MathSciNetView ArticleGoogle Scholar
- Mohiuddine SA, Alotaibi A: Statistical convergence and approximation theorems for functions of two variables. J. Comput. Anal. Appl. 2013, 15(2):218–223.MathSciNetGoogle Scholar
- Mohiuddine SA, Alotaibi A, Mursaleen M:Statistical summability and a Korovkin type approximation theorem. J. Inequal. Appl. 2012., 2012: Article ID 172Google Scholar
- Mursaleen M, Alotaibi A: Statistical summability and approximation by de la Vallée-Poussin mean. Appl. Math. Lett. 2011, 24: 320–324. 10.1016/j.aml.2010.10.014MathSciNetView ArticleGoogle Scholar
- Mursaleen M, Alotaibi A: Statistical lacunary summability and a Korovkin type approximation theorem. Ann. Univ. Ferrara 2011, 57(2):373–381. 10.1007/s11565-011-0122-8MathSciNetView ArticleGoogle Scholar
- Mursaleen M, Karakaya V, Ertürk M, Gürsoy F: Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput. 2012, 218: 9132–9137. 10.1016/j.amc.2012.02.068MathSciNetView ArticleGoogle Scholar
- Srivastava HM, Mursaleen M, Khan A: Generalized equi-statistical convergence of positive linear operators and associated approximation theorems. Math. Comput. Model. 2012, 55: 2040–2051. 10.1016/j.mcm.2011.12.011MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.