- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 370271, 7 pages
On Generalized Statistical Convergence of Order of Difference Sequences
Department of Mathematics, Fırat University, 23119 Elazığ, Turkey
Received 31 May 2013; Accepted 16 August 2013
Academic Editor: Mihail Megan
Copyright © 2013 Mikail Et et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We introduce the concept of statistical convergence of order of difference sequences, and we give some relations between the set of statistical convergence of order of difference sequences and strong -summability of order . Furthermore some relations between the spaces and are examined.
The idea of statistical convergence was given by Zygmund  in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Steinhaus  and Fast  and later reintroduced by Schoenberg  independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory, and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Connor , Fridy , Khan and Orhan , Miller and Orhan , Mursaleen , Rath and Tripathy , Šalát , Savaş , and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Čech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability.
Let be the set of all sequences of real or complex numbers, and let , , and be, respectively, the Banach spaces of bounded, convergent, and null sequences with the usual norm , where , the set of positive integers. Also by , , , and , we denote the spaces of all bounded, convergent, absolutely, and -absolutely convergent series, respectively.
Let be a nondecreasing sequence of positive numbers tending to such that , . The generalized de la Vallée-Poussin mean is defined by , where for . A sequence is said to be -summable to a number if as . If , then -summability is reduced to the Cesàro summability.
The notion of difference sequence spaces was introduced by Kızmaz , and it was generalized by Et and Çolak  such as for , , or , where , , , and so . The sequence spaces are the Banach spaces normed by for , or . Recently the difference sequence spaces have been studied in [16–21].
The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan  and after that statistical convergence of order and strongly -Cesàro summability of order were studied by Çolak  and generalized by Çolak and Bektaş .
Let be given. The sequence is said to be statistically convergent of order if there is a real number such that for every . In this case we write . The set of all statistically convergent sequences of order will be denoted by .
Let be a sequence space. Then is called(i)solid (or normal) if whenever for all sequences of scalars with ,(ii)symmetric if implies , where is a permutation of ,(iii)sequence algebra if is closed under multiplication.
2. Main Results
In this section we give the main results of the paper. In Theorem 5 we give the inclusion relations between the sets of -statistically convergent sequences of order for different ’s. In Theorem 9 we give the relationship between the strong -summability of order and the strong -summability of order . In Theorem 12 we give the relationship between the strong -summability of order and the -statistical convergence of order .
Definition 1. Let the sequence of real numbers be defined as above, and let be given. The sequence is said to be -statistically convergent of order if there is a real number such that where and denote the th power of ; that is . In this case we write . The set of all -statistically convergent sequences of order will be denoted by . For for all , we will write instead of and in the special case , we will write instead of , and also in the special case and for all we will write instead of , which were defined and studied by Et and Nuray .
The -statistical convergence of order is well defined for , but it is not well defined for in general. For this let for all and be defined as follows: and so then both for , and so -statistically converges of order , both to and ; that is, and . But this is impossible.
Theorem 2. Let and , be sequences of real numbers; then(i)if and , then ,(ii)if and , then .
Definition 3. Let the sequence be as above, any real number, and a positive real number. A sequence is said to be strongly -summable of order if there is a real number such that where . In this case we write . The strong -summability of order reduces to the strong -summability for . The set of all strongly -summable sequences of order will be denoted by .
Theorem 4. Let the sequence be as above, , and a positive real number. The sequence space is a Banach space for normed by and a complete -normed space for by
Proof. It is a routine verification that is a normed space normed by (9). Let be a Cauchy sequence such that for each . Then we have
as and so that
as . Now for we get as , and for all we get as . Since the inequality
holds for all , we get as and for each . This implies that the sequence is a Cauchy in for each . Also, it is convergent, because is complete. Assume that for each . Since is a Cauchy sequence, then for every there exists a number such that
for all . Hence for all and all we get
Taking limit as in the last inequalities, we get
for and all . This implies that
and hence as . Since
we get .
It can be shown that is a complete -normed space for by (10).
Theorem 5. If , then and the inclusion is strict.
Proof. Proof follows from the inequality To show that the inclusion is strict, define such that Then for , but for .
From Theorem 5 we have the following.
Corollary 6. If a sequence is -statistically convergent of order , to , then it is -statistically convergent to .
Theorem 7. if
Proof. For given we have and so Taking the limit as and using (21), we get .
Theorem 8. if and only if
Theorem 9. Let and a positive real number; then and the inclusion is strict.
Proof. Let . Then given and such that and a positive real number , we may write
and this gives that .
To show that the inclusion is strict define such that Then for but for , for .
The following result is a consequence of Theorem 9.
Corollary 10. Let and a positive real number. Then(i)if , then ,(ii) for each and .
Theorem 11. Let and . Then .
Proof is seen from the Hölder inequality.
Theorem 12. Let and be fixed real numbers such that and . If a sequence is strongly -summable of order , to , then it is -statistically convergent of order , to .
Proof. For any sequence and , we have and so From this it follows that if is strongly -summable of order , to , then it is -statistically convergent of order , to .
Corollary 13. Let be a fixed real number such that and .(i)If a sequence is strongly -summable of order , to , then it is -statistically convergent of order , to .(ii)If a sequence is strongly -summable of order , to , then it is -statistically convergent to .
3. Results Related to the Orlicz Function
In this section we give the inclusion relations between the sets of -statistically convergent sequences of order and strongly -summable sequences of order with respect to the Orlicz function .
An Orlicz function is a function , which is continuous, nondecreasing, and convex with , for and as . An Orlicz function can always be represented in the following integral form: where known as the kernel of is right differentiable for , , for , is nondecreasing, and as .
Let be a finite measure space. We denote by the space of all (equivalence classes of) -measurable functions from into . Given an Orlicz function , we define on a convex functional by and an Orlicz space by (for detail see ).
The study of Orlicz sequence spaces was initiated with a certain specific purpose in the Banach space theory. Indeed, Lindberg  got interested in Orlicz spaces in connection with finding the Banach spaces with symmetric Schauder bases having complementary subspaces isomorphic to or . Subsequently, Lindenstrauss and Tzafriri  used the idea of the Orlicz function to construct the sequence space The space with the norm becomes a Banach space, called an Orlicz sequence space. The space is closely related to the space which is an Orlicz sequence space with for . Lindenstrauss and Tzafriri  proved that every Orlicz sequence space contains a subspace isomorphic to . The Orlicz sequence spaces are the special cases of Orlicz spaces studied in .
It is well known that if is a convex function and , then for all with .
Definition 14. Let an Orlicz function, a sequence of strictly positive real numbers, and any real number. Now we define If , then we say that is strongly -summable of order with respect to the Orlicz function . For for all and we will write instead of and in the special case we will write instead of .
In the following two theorems we will assume that the sequence is bounded and .
Theorem 15. Let be real numbers such that and an Orlicz function; then.
Proof. Let , be given, and let and denote the sums over , and , , respectively. Since for each we may write Since , the left-hand side of the above inequality tends to zero as . Therefore the right-hand side tends to zero as and hence , because .
Corollary 16. Let and an Orlicz function; then .
Theorem 17. Let be an Orlicz function and -bounded sequence; then .
Proof. Suppose that a -bounded sequence; that is, and . Since , then there is a constant such that . Given we have Hence .
Theorem 18. The sequence spaces , , and are neither solid nor symmetric, nor sequence algebras for .
Proof. Let for all ; then but when for all . Hence is not solid. The other cases can be proved on considering similar examples.
Theorem 19. If and is strongly -summable of order , to with respect to the Orlicz function , then uniquely.
Proof. Let . Suppose that and . Then
Define . Since is nondecreasing and convex, we have where and . Hence Since we have and so . Thus the limit is unique.
- A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, UK, 1979.
- H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloquium Mathematicum, vol. 2, pp. 73–74, 1951.
- H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951.
- I. J. Schoenberg, “The integrability of certain functions and related summability methods,” The American Mathematical Monthly, vol. 66, pp. 361–375, 1959.
- J. S. Connor, “The statistical and strong -Cesàro convergence of sequences,” Analysis, vol. 8, no. 1-2, pp. 47–63, 1988.
- J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985.
- M. K. Khan and C. Orhan, “Characterizations of strong and statistical convergences,” Publicationes Mathematicae Debrecen, vol. 76, no. 1-2, pp. 77–88, 2010.
- H. I. Miller and C. Orhan, “On almost convergent and statistically convergent subsequences,” Acta Mathematica Hungarica, vol. 93, no. 1-2, pp. 135–151, 2001.
- M. Mursaleen, “-statistical convergence,” Mathematica Slovaca, vol. 50, no. 1, pp. 111–115, 2000.
- D. Rath and B. C. Tripathy, “On statistically convergent and statistically Cauchy sequences,” Indian Journal of Pure and Applied Mathematics, vol. 25, no. 4, pp. 381–386, 1994.
- T. Šalát, “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, no. 2, pp. 139–150, 1980.
- E. Savaş, “Strong almost convergence and almost -statistical convergence,” Hokkaido Mathematical Journal, vol. 29, no. 3, pp. 531–536, 2000.
- L. Leindler, “Über die verallgemeinerte de la Vallée-Poussinsche Summierbarkeit allgemeiner Orthogonalreihen,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16, pp. 375–387, 1965.
- H. Kızmaz, “On certain sequence spaces,” Canadian Mathematical Bulletin, vol. 24, no. 2, pp. 169–176, 1981.
- M. Et and R. Çolak, “On some generalized difference sequence spaces,” Soochow Journal of Mathematics, vol. 21, no. 4, pp. 377–386, 1995.
- B. Altay and F. Başar, “On the fine spectrum of the difference operator on and ,” Information Sciences, vol. 168, no. 1–4, pp. 217–224, 2004.
- Ç. A. Bektaş, M. Et, and R. Çolak, “Generalized difference sequence spaces and their dual spaces,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 423–432, 2004.
- N. L. Braha, “A new class of sequences related to the spaces defined by sequences of Orlicz functions,” Journal of Inequalities and Applications, vol. 2011, Article ID 539745, 10 pages, 2011.
- M. Et, “Spaces of Cesàro difference sequences of order defined by a modulus function in a locally convex space,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 865–879, 2006.
- M. Işik, “Generalized vector-valued sequence spaces defined by modulus functions,” Journal of Inequalities and Applications, vol. 2010, Article ID 457892, 7 pages, 2010.
- B. C. Tripathy, Y. Altin, and M. Et, “Generalized difference sequence spaces on seminormed space defined by Orlicz functions,” Mathematica Slovaca, vol. 58, no. 3, pp. 315–324, 2008.
- A. D. Gadjiev and C. Orhan, “Some approximation theorems via statistical convergence,” The Rocky Mountain Journal of Mathematics, vol. 32, no. 1, pp. 129–138, 2002.
- R. Çolak, “Statistical convergence of order α,” in Modern Methods in Analysis and Its Applications, pp. 121–129, Anamaya, New Delhi, India, 2010.
- R. Çolak and Ç. A. Bektaş, “-statistical convergence of order ,” Acta Mathematica Scientia. Series B, vol. 31, no. 3, pp. 953–959, 2011.
- M. Et and F. Nuray, “-statistical convergence,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 6, pp. 961–969, 2001.
- M. A. Krasnoselskiĭ and J. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Gorningen, Netherlands, 1961.
- K. Lindberg, “On subspaces of Orlicz sequence spaces,” Studia Mathematica, vol. 45, pp. 119–146, 1973.
- J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces,” Israel Journal of Mathematics, vol. 10, pp. 379–390, 1971.
- Y. Altin, M. Et, and B. C. Tripathy, “The sequence space on seminormed spaces,” Applied Mathematics and Computation, vol. 154, no. 2, pp. 423–430, 2004.
- V. K. Bhardwaj and N. Singh, “Some sequence spaces defined by Orlicz functions,” Demonstratio Mathematica, vol. 33, no. 3, pp. 571–582, 2000.
- M. Et, Y. Altin, B. Choudhary, and B. C. Tripathy, “On some classes of sequences defined by sequences of Orlicz functions,” Mathematical Inequalities & Applications, vol. 9, no. 2, pp. 335–342, 2006.
- M. Güngör, M. Et, and Y. Altin, “Strongly -summable sequences defined by Orlicz functions,” Applied Mathematics and Computation, vol. 157, no. 2, pp. 561–571, 2004.
- M. Mursaleen, Q. A. Khan, and T. A. Chishti, “Some new convergent sequences spaces defined by Orlicz functions and statistical convergence,” Italian Journal of Pure and Applied Mathematics, no. 9, pp. 25–32, 2001.
- E. Savaş and B. E. Rhoades, “On some new sequence spaces of invariant means defined by Orlicz functions,” Mathematical Inequalities & Applications, vol. 5, no. 2, pp. 271–281, 2002.