Skip to main content
SpringerLink
Log in

Almost and Statistical Convergence of Ordinary Sequences: A Preview

  • Chapter
Convergence Methods for Double Sequences and Applications

  • 743 Accesses

Abstract

In this chapter, we recall the notion of almost convergence and statistical convergence for single sequences x=(x k ). We present here a brief survey on developments of almost convergence, statistical convergence, and some related methods, e.g., absolute almost convergence and strong almost convergence for single sequences.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Z.U. Ahmad, M. Mursaleen, An application of Banach limits. Proc. Am. Math. Soc. 103(1), 244–246 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Banach, Théorie des operations liniaries (Warszawa, 1932)

    Google Scholar 

  3. F. BaÅŸar, Summability Theory and Its Applications (Bentham Science Publishers, Istanbul, 2011)

    Google Scholar 

  4. F. Başar, M. Kirişçi, Almost convergence and generalized difference matrix. Comput. Math. Appl. 61, 602–611 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Das, Banach and other limits. J. Lond. Math. Soc. (2) 7, 327–347 (1973)

    Google Scholar 

  6. G. Das, B. Kuttner, Space of absolute almost convergence. Indian J. Math. 28(3), 241–257 (1986)

    MathSciNet  Google Scholar 

  7. G. Das, S.K. Mishra, A note on a theorem of Maddox on strong almost convergence. Math. Proc. Camb. Philos. Soc. 89, 393–396 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Das, S.K. Sahoo, A generalization of strong and absolute almost convergence. J. Indian Math. Soc. 58, 65–74 (1992)

    MathSciNet  MATH  Google Scholar 

  9. G. Das, S.K. Sahoo, On some sequence spaces. J. Math. Anal. Appl. 164, 381–398 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Das, B. Kuttner, S. Nanda, Some sequence spaces and absolute almost convergence. Trans. Am. Math. Soc. 283, 729–739 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Das, B. Kuttner, S. Nanda, On absolute almost convergence. J. Math. Anal. Appl. 164, 381–398 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. J.P. Duran, Infinite matrices and almost convergence. Math. Z. 128, 75–83 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. J.P. Duran, Almost convergence, summability and ergodicity. Can. J. Math. 26(2), 372–387 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Eizen, G. Laush, Infinite matrices and almost convergence. Math. Jpn. 14, 137–143 (1969)

    MathSciNet  MATH  Google Scholar 

  15. H. Fast, Surla convergence statistique. Colloq. Math. 2, 241–244 (1951)

    MathSciNet  MATH  Google Scholar 

  16. A.P. Freedman, J.J. Sember, Densities and summability. Pac. J. Math. 95, 293–305 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.A. Fridy, Statistical limit points. Proc. Am. Math. Soc. 118, 1187–1192 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. J.P. King, Almost summable sequences. Proc. Am. Math. Soc. 17, 1219–1225 (1966)

    Article  MATH  Google Scholar 

  20. K. Knopp, Zur Theorie der Limitierungsverfahren (Erste Mitteilung). Math. Z. 31, 115–127 (1930)

    Google Scholar 

  21. E. Kolk, The statistica convergence in Banach spaces. Tartu Ülik Toimetised 928, 41–52 (1991)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. I.J. Maddox, A new type of convergence. Math. Proc. Camb. Philos. Soc. 83, 61–64 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  24. I.J. Maddox, On strong almost convergence. Math. Proc. Camb. Philos. Soc. 85, 345–350 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Mursaleen, On some new invariant matrix methods of summability. Q. J. Math. (Oxford) 34, 77–86 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Mursaleen, Matrix transformations between some new sequence spaces. Houst. J. Math. 9, 505–509 (1983)

    MathSciNet  MATH  Google Scholar 

  27. M. Mursaleen, Infinite matrices and absolute almost convergence. Int. J. Math. Math. Sci. 6, 503–510 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Mursaleen, Absolute almost convergent sequences. Houst. J. Math. 10(3), 427–431 (1984)

    MathSciNet  MATH  Google Scholar 

  29. M. Mursaleen, Some matrix transformations on sequence spaces of invariant means. Hacet. J. Math. Stat. 38(3), 259–264 (2009)

    MathSciNet  MATH  Google Scholar 

  30. M. Mursaleen, On \(\mathcal{A}\)-invariant mean and \(\mathcal{A}\)-almost convergence. Anal. Math. 37(3), 173–180 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. R.A. Raimi, Invariant means and invariant matrix methods of summability. Duke Math. J. 30, 81–94 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Šalát, On statistically convergent sequences of real numbers. Math. Slovaca 30, 139–150 (1980)

    MathSciNet  MATH  Google Scholar 

  33. P. Schaefer, Infinite matrices and invariant means. Proc. Am. Math. Soc. 36, 104–110 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  34. I.J. Schoenberg, The integrability of certain functions and related summability methods. Am. Math. Mon. 66, 361–375 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  35. S. Simons, The sequence spaces l(p ν ) and m(p ν ). Proc. Lond. Math. Soc. (3) 15, 422–436 (1965)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  38. M. Stieglitz, Eine Verallgemeinerung des Begriffs der Fastkonvergenz. Math. Jpn. 18, 53–70 (1973)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, Aligarh, Uttar Pradesh, India

    M. Mursaleen

  2. Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

    S. A. Mohiuddine

Authors
  1. M. Mursaleen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. A. Mohiuddine
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer India

About this chapter

Cite this chapter

Mursaleen, M., Mohiuddine, S.A. (2014). Almost and Statistical Convergence of Ordinary Sequences: A Preview. In: Convergence Methods for Double Sequences and Applications. Springer, New Delhi. https://doi.org/10.1007/978-81-322-1611-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation