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\(A\)-distributional summability in topological spaces

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Abstract

In the present paper we introduce a new concept of \(A\)-distributional convergence in an arbitrary Hausdorff topological space which is equivalent to \(A\)-statistical convergence for a degenerate distribution function. We investigate \(A\)-distributional convergence as a summability method in an arbitrary Hausdorff topological space. We also study the summability of spliced sequences, in particular, for metric spaces and give the Bochner integral representation of \(A\)-limits of the spliced sequences for Banach spaces.

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References

  1. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    MATH  Google Scholar 

  2. Buck, R.C.: Generalized asymptotic density. Am. J. Math. 75, 335–346 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  3. Buck, R.C.: The measure theoretic approach to density. Am. J. Math. 68, 560–580 (1946)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cakalli, H., Khan, M.K.: Summability in topological spaces. Appl. Math. Lett. 24(3), 348–352 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cakalli, H.: Lacunary statistical convergence in topological groups. Indian J. Pure Appl. Math. 26(2), 113–119 (1995)

    MATH  MathSciNet  Google Scholar 

  6. Cakalli, H.: On statistical convergence in topological groups. Pure Appl. Math. Sci. 43(1–2), 27–31 (1996)

    MATH  MathSciNet  Google Scholar 

  7. Cesàro, E.: Sur la multiplication des séries. Bulletin des Sciences Mathématiques. 2, 114–120 (1890)

    Google Scholar 

  8. Connor, J.S.: The statistical and strong p-Cesàro convergence of sequences. Analysis 8(1–2), 47–63 (1988)

    MATH  MathSciNet  Google Scholar 

  9. Di Maio, G., Kočinac, L.D.R.: Statistical convergence in topology. Topol. Appl. 156(1), 28–45 (2008)

    Article  MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  11. Fridy, J.A.: On statistical convergence. Analysis 5(4), 301–313 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fridy, J.A., Miller, H.I.: A matrix characterization of statistical convergence. Analysis 11(1), 59–66 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. Khan, M.K., Orhan, C.: Matrix characterization of A-statistical convergence. J. Math. Anal. Appl. 335(1), 406–417 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Khan, M.K., Orhan, C.: Characterization of strong and statistical convergences. Publ. Math. Debrecen 76(1–2), 77–88 (2010)

    MATH  MathSciNet  Google Scholar 

  15. Osikiewicz, J.A.: Summability of spliced sequences. Rocky Mountain J. Math. 35(3), 977–996 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Powell, R.E., Shah, S.M.: Summability Theory and Its Applications. Prentice-Hall of India, New Delhi (1988)

    Google Scholar 

  17. Prullage, D.L.: Summability in topological groups. Math. Z. 96, 259–278 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  18. Prullage, D.L.: Summability in topological groups, II. Math. Z. 103, 129–138 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  19. Prullage, D.L.: Summability in topological groups, IV. Convergence fields. Tôhoku Math. J. 21(2), 159–169 (1969)

    Google Scholar 

  20. Prullage, D.L.: Summability in topological groups. III. Metric properties. J. Analyse Math. 22, 221–231 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  21. Šalát, T.: On statistically convergent sequences of real numbers. Math. Slovaca 30(2), 139–150 (1980)

    MATH  MathSciNet  Google Scholar 

  22. Zygmund, A.: Trigonometric Series, vol. 1. Cambridge University Press, Cambridge (1959)

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Authors and Affiliations

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

    M. Unver & C. Orhan

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

    M. Khan

Authors
  1. M. Unver
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  2. M. Khan
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  3. C. Orhan
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Corresponding author

Correspondence to M. Unver.

Additional information

This research was done when the first author was visiting Kent State University and the research was supported by the Higher Education Council of Turkey (YÖK)

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Unver, M., Khan, M. & Orhan, C. \(A\)-distributional summability in topological spaces. Positivity 18, 131–145 (2014). https://doi.org/10.1007/s11117-013-0235-7

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  • DOI: https://doi.org/10.1007/s11117-013-0235-7

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