Skip to main content
SpringerLink
Log in

Generalized Limits and Statistical Convergence

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

Consider the Banach space m of real bounded sequences, x, with \({\Vert x\Vert =\sup_{k}|x_{k}|}\). A positive linear functional L on m is called an S-limit if \({L(\chi _{K})=0}\) for every characteristic sequence \({\chi _{K} }\) of sets, K, of natural density zero. We provide regular sublinear functionals that both generate as well as dominate S-limits. The paper also shows that the set of S-limits and the collection of Banach limits are distinct but their intersection is not empty. Furthermore, we show that the generalized limits generated by translative regular methods is equal to the set of Banach limits. Some applications are also provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agnew R.P.: A simple sufficient condition that a method of summability be stronger than convergence. Bull. Am. Math. Soc. 52, 128–132 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  2. Banach, S.: Théorie des Opérations Linéares. Warsaw (1932)

  3. Bennett G., Kalton N.J.: Consistency theorems for almost convergence. Trans. Am. Math. Soc. 198, 23–43 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boos, J.: Classical and Modern Methods in Summability. Oxford University Press, Oxford, (2000)

  5. Chang, C.-T., Chen, C.-P.: Matrix maps of statistically convergent sequences. Linear Algebra Appl. 437, 2896–2909 (2012)

  6. Connor J.: Two valued measures and summability. Analysis 10(4), 373–385 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Connor, J.: A topological and functional analytic approach to statistical convergence. In: Analysis of Divergence, Orono, ME, (1997), in: Appl. Numer. Anal., Birkhäuser, Boston, MA, 403–413 (1999)

  8. Connor J., Fridy J.A., Orhan C.: Core equality results for sequences. J. Math. Anal. Appl. 321, 515–523 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Demirci K.: A-statistical core of a sequence. Demonstr. Math. 33(2), 343–353 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Demirci K., Khan M.K., Orhan C.: Strong and A-statistical comparisons for sequences. J. Math. Anal. Appl. 278, 27–33 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Duran P.J.: Strongly regular matrices, almost convergence and Banach limits. Duke M. J. 39, 497–502 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fast H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)

    MathSciNet  MATH  Google Scholar 

  13. Freedman A.R.: Generalized limits and sequence spaces. Bull. Lond. Math. Soc. 13, 224–228 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fridy J.A.: On statistical convergence. Analysis 5, 301–313 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fridy J.A., Miller H.I.: A matrix characterization of statistical convergence. Analysis 11, 59–66 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fridy J.A., Orhan C.: Statistical limit superior and limit inferior. Proc. Am. Math. Soc. 125(12), 3625–3631 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goffman, C., Pedrick, G.: First Course in Functional Analysis. Chelsea Publishing Company, New York (1983)

  18. Hill J.D.: Remarks on the Borel property. Pac. J. Math. 4, 227–242 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jerison M.: The set of all generalized limits of bounded sequences. Canad. J. Math. 9, 79–89 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  20. Khan M.K., Orhan C.: Matrix characterization of A-statistical convergence. J. Math. Anal. Appl. 335, 406–417 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lorentz G.G.: A contribution to the theory of divergent sequences. Acta Math. 80, 167–190 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  22. Miller H.I.: A measure theoretical subsequence characterization of statistical convergence. Trans. Am. Math. Soc. 347, 1811–1819 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Miller H.I., Orhan C.: On almost convergent and statistically convergent subsequences. Acta Math. Hung. 93(1-2), 135–151 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Osikiewicz J.A.: Summability of spliced sequences. Rocky Mountain J. Math. 35, 977–996 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Petersen, G.M.: Regular Matrix Transformations. McGraw Hill, London (1966)

  26. Simons S.: Banach limits, infinite matrices and sublinear functionals. J. Math. Anal. Appl. 26, 640–655 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  27. Steinhaus H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2, 73–74 (1951)

    MathSciNet  Google Scholar 

  28. Unver M., Khan M.K., Orhan C.: A-distributional summability in topological spaces. Positivity 18(1), 131–145 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Science, Department of Mathematics, Ankara University, Tandoğan, 06100, Ankara, Turkey

    T. Yurdakadim & C. Orhan

  2. Department of Mathematics, Kent State University, Kent, OH, 44242, USA

    M. K. Khan

  3. Faculty of Engineering and Natural Sciences, International University of Sarajevo, Sarajevo, 71000, Bosnia-Herzegovina

    H. I. Miller

Authors
  1. T. Yurdakadim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. K. Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. I. Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Orhan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Yurdakadim.

Additional information

A part of this research was conducted when the first author was visiting Kent State University. This research was supported by the Higher Education Council of Turkey (YÖK).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yurdakadim, T., Khan, M.K., Miller, H.I. et al. Generalized Limits and Statistical Convergence. Mediterr. J. Math. 13, 1135–1149 (2016). https://doi.org/10.1007/s00009-015-0554-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-015-0554-y

Mathematics Subject Classification

Keywords

Search

Navigation