Banach Journal of Mathematical Analysis

The sequence space $E_{n}^{q}\left( M,p,s\right) $ and $ N_{k}-$ lacunary statistical convergence

Naim L. Braha and Mikail Et

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Abstract

In this paper we define the sequence space $E_{n}^{q}(M,p,s)$ by using an Orlicz function and we study various properties and obtain some inclusion relations involving this space. We give some relations between $N_{k}-$lacunary statistical convergence and strongly $N_{k}-$lacunary convergence.

Article information

Source
Banach J. Math. Anal., Volume 7, Number 1 (2013), 88-96 .

Dates
First available in Project Euclid: 22 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1358864550

Digital Object Identifier
doi:10.15352/bjma/1358864550

Mathematical Reviews number (MathSciNet)
MR3004268

Zentralblatt MATH identifier
1270.46005

Subjects
Primary: 40A05: Convergence and divergence of series and sequences
Secondary: 40C05: Matrix methods 46A45: Sequence spaces (including Köthe sequence spaces) [See also 46B45]

Keywords
Orlicz function sequence space Euler mean statistical convergence

Citation

Braha, Naim L.; Et, Mikail. The sequence space $E_{n}^{q}\left( M,p,s\right) $ and $ N_{k}-$ lacunary statistical convergence. Banach J. Math. Anal. 7 (2013), no. 1, 88--96. doi:10.15352/bjma/1358864550. https://projecteuclid.org/euclid.bjma/1358864550


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References

  • Y. Alt\i n, M. Et and B.C. Tripathy, The sequence space $|\overline{N}\sb p|(M,r,q,s)$ on seminormed spaces, Appl. Math. Comput. 154 (2004), no. 2, 423–430.
  • Y. Altun and T. Bilgin, On a new class of sequences related to the $l_{p}$ space defined by Orlicz function, Taiwanese J. Math. 13 (2009), no. 4, 1189–1196.
  • V.K. Bhardwaj and N. Singh, Some sequence spaces defined by Orlicz functions, Demonstratio Math. 33 (2000), no. 3, 571–582.
  • J.S. Connor, The Statistical and strong $p$-Cesaro convergence of sequences, Analysis 8 (1988), 47–63.
  • R. Çolak, Statistical convergence of order $\alpha ,$ Modern Methods in Analysis and its Applications, Anamaya Publishers, New Delhi, India, 2010.
  • R. Çolak, B.C. Tripathy and M. Et, Lacunary strongly summable sequences and $q-$lacunary almost statistical convergence, Vietnam J. Math. 34 (2006), no. 2, 129–138.
  • N.L. Braha, A new class of sequences related to the $l_{p}$ spaces defined by sequences of Orlicz functions, J. Inequal. Appl. 2011, Article ID 539745, 10 p. (2011).
  • G. Das and S.K. Mishra, Banach limits and lacunary strong almost convegence, J. Orissa Math. Soc. 2 (1983), 61–70.
  • M. Et, Y. Altin, Choudhary, B. and B.C. Tripathy, On some classes of sequences defined by sequences of Orlicz functions, Math. Inequal. Appl. 9 (2006), no. 2, 335–342.
  • M. Et, A. Gökhan and H. Altinok, On statistical convergence of vector-valued sequences associated with multiplier sequences, Ukrainian Math. J. 58 (2006), no. 1, 139–146
  • H. Fast, Sur la convergence statistique, Colloquium Math. 2 (1951), 241–244.
  • A.R. Freedman, J.J. Sember and M. Raphael, Some Cesaro-type summability spaces, Proc. London Math. Soc. 37 (1978), 508–520.
  • J.A. Fridy, On the statistical convergence, Analysis 5 (1985), 301– 313.
  • J.A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993), 43–51.
  • P.K. Kamptan and M. Gupta, Sequence Spaces and Series, Marcel Dekker Inc., New York, 1981.
  • M.A. Krasnoselskii and Y.B. Rutitsky, Convex function and Orlicz spaces, Groningen, Netherland, 1961.
  • J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math. 1 (1971), 379–390.
  • M. Mursaleen, Q.A. Khan and T.A. Chishti, Some new convergent sequences spaces defined by Orlicz functions and statistical convergence, Ital. J. Pure Appl. Math. 9 (2001), 25–32.
  • M. Mursaleen, $\lambda -$ statistical convergence, Math. Slovaca 50 (2000), no. 1, 111–115.
  • D. Rath and B.C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian J. Pure. Appl. Math., 25 (1994), no. 4, 381–386.
  • T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139–150.
  • E. Savaş, Strong almost convergence and almost $ \lambda $-statistically convergence, Hokkaido Math. J. 29 (2000), 531-536.
  • E. Savas and B.E. Rhoades, On some new sequence spaces of invariant means defined by Orlicz functions, Math. Inequal. Appl. 5 (2002), no. 2, 271–281.
  • I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375.
  • H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloquium Math. 2 (1951), 73–74.
  • A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, UK, 1979.