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On statistical convergence of vector-valued sequences associated with multiplier sequences

  • M. Et
  • A. Gökhan
  • H. Altinok
Article

Abstract

We introduce vector-valued sequence spaces w (F, Q, p, u), w 1(F, Q, p, u), w 0(F, Q, p, u), S u q , and S 0u q using a sequence of modulus functions and a multiplier sequence u = (u k ) of nonzero complex numbers. We give some relations for these sequence spaces. It is also shown that if a sequence is strongly u q -Cesàro summable with respect to the modulus function, then it is u q -statistically convergent.

Keywords

Complex Number Sequence Space Statistical Convergence Modulus Function Nonzero Complex Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    A. Rath and P. D. Srivastava, “On some vector valued sequence spaces ℓ(p) (E k, Λ),” Ganita, 47, No. 1, 1–12 (1996).MathSciNetGoogle Scholar
  2. 2.
    N. R. Das and A. Choudhary, “Matrix transformation of vector valued sequence spaces,” Bull. Calcutta Math. Soc., 84, 47–54 (1992).MathSciNetzbMATHGoogle Scholar
  3. 3.
    I. E. Leonard, “Banach sequence spaces,” J. Math. Anal. Appl., 54, 245–265 (1976).zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    J. K. Srivastava and B. K. Srivastava, “Generalized sequence space c 0(X, λ, p),” Indian J. Pure Appl. Math., 27, No. 1, 73–84 (1996).MathSciNetzbMATHGoogle Scholar
  5. 5.
    B. C. Tripathy and M. Sen, “Vector valued paranormed bounded and null sequence spaces associated with multiplier sequences,” Soochow J. Math., 29, No. 4, 379–391 (2003).MathSciNetzbMATHGoogle Scholar
  6. 6.
    B. C. Tripathy and S. Mahanta, “On a class of vector valued sequences associated with multiplier sequences,” Acta Math. Appl. Sinica (to appear).Google Scholar
  7. 7.
    R. Çolak, “On invariant sequence spaces,” Erc. Univ. J. Sci., 5, No. 1–2, 881–887 (1989).Google Scholar
  8. 8.
    R. Çolak, P. D. Srivastava, and S. Nanda, “On certain sequence spaces and their Köthe-Toeplitz duals,” Rend. Mat. Appl., Ser 13, No. 1, 27–39 (1993).Google Scholar
  9. 9.
    H. Nakano, “Concave modulars,” J. Math. Soc. Jpn., 5, 29–49 (1953).zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    W. H. Ruckle, “FK spaces in which the sequence of coordinate vectors is bounded,” Can. J. Math., 25, 973–978 (1973).zbMATHMathSciNetGoogle Scholar
  11. 11.
    I. J. Maddox, “Sequence spaces defined by a modulus,” Math. Proc. Cambridge Phil. Soc., 100, 161–166 (1986).zbMATHMathSciNetGoogle Scholar
  12. 12.
    I. J. Maddox, “On strong almost convergence,” Math. Proc. Cambridge Phil. Soc., 85, 161–166 (1979).MathSciNetGoogle Scholar
  13. 13.
    H. Fast, “Sur la convergence statistique,” Colloq. Math., 2, 241–244 (1951).zbMATHMathSciNetGoogle Scholar
  14. 14.
    I. J. Schoenberg, “The integrability of certain functions and related summability methods,” Amer. Math. Monthly, 66, 361–375 (1959).zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    J. A. Fridy, “On the statistical convergence,” Analysis, 5, 301–313 (1985).zbMATHMathSciNetGoogle Scholar
  16. 16.
    J. S. Connor, “A topological and functional analytic approach to statistical convergence,” Appl. Numer. Harmonic Anal., 403–413 (1999).Google Scholar
  17. 17.
    T. Šalàt, “On statistically convergent sequences of real numbers,” Math. Slovaca, 30, No. 2, 139–150 (1980).MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mursaleen, “λ-Statistical convergence,” Math. Slovaca, 50, 111–115 (2000).MathSciNetzbMATHGoogle Scholar
  19. 19.
    M. Isik, “On statistical convergence of generalized difference sequences,” Soochow J. Math., 30, No. 2, 197–205 (2004).zbMATHMathSciNetGoogle Scholar
  20. 20.
    E. Savaş, “Strong almost convergence and almost λ-statistical convergence,” Hokkaido Math. J., 29, 531–536 (2000).MathSciNetzbMATHGoogle Scholar
  21. 21.
    E. Malkowsky and E. Savaş, “Some λ-sequence spaces defined by a modulus,” Arch. Math., 36, 219–228 (2000).zbMATHGoogle Scholar
  22. 22.
    E. Kolk, “The statistical convergence in Banach spaces,” Acta. Comment. Univ. Tartu, 928, 41–52 (1991).MathSciNetGoogle Scholar
  23. 23.
    I. J. Maddox, “Statistical convergence in a locally convex space,” Math. Proc. Cambridge Phil. Soc., 104, 141–145 (1988).zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    B. C. Tripathy and M. Sen, “On generalized statistically convergent sequences,” Indian J. Pure Appl. Math., 32, No. 11, 1689–1694 (2001).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • M. Et
    • 1
  • A. Gökhan
    • 1
  • H. Altinok
    • 1
  1. 1.Fırat UniversityElaziğgTurkey

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