International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2007 / Article

Research Article | Open Access

Volume 2007 |Article ID 038530 | https://doi.org/10.1155/2007/38530

Vinod K. Bhardwaj, Indu Bala, "On Weak Statistical Convergence", International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 038530, 9 pages, 2007. https://doi.org/10.1155/2007/38530

On Weak Statistical Convergence

Academic Editor: Narendra K. Govil
Received27 Jun 2007
Revised28 Sep 2007
Accepted10 Oct 2007
Published01 Jan 2008

Abstract

The main object of this paper is to introduce a new concept of weak statistically Cauchy sequence in a normed space. It is shown that in a reflexive space, weak statistically Cauchy sequences are the same as weakly statistically convergent sequences. Finally, weak statistical convergence has been discussed in lp spaces.

References

  1. A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, UK, 1979.
  2. H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloquium Mathematicum, vol. 2, pp. 73–74, 1951. View at: Google Scholar
  3. H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  4. I. J. Schoenberg, “The integrability of certain functions and related summability methods,” The American Mathematical Monthly, vol. 66, no. 5, pp. 361–375, 1959. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. P. Erdös and G. Tenenbaum, “Sur les densités de certaines suites d'entiers,” Proceedings of the London Mathematical Society, vol. 59, no. 3, pp. 417–438, 1989. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  6. H. I. Miller, “A measure theoretical subsequence characterization of statistical convergence,” Transactions of the American Mathematical Society, vol. 347, no. 5, pp. 1811–1819, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. A. R. Freedman and J. J. Sember, “Densities and summability,” Pacific Journal of Mathematics, vol. 95, no. 2, pp. 293–305, 1981. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  8. I. J. Maddox, “Statistical convergence in a locally convex space,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 1, pp. 141–145, 1988. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  9. J. Connor and M. A. Swardson, “Strong integral summability and the Stone-Čech compactification of the half-line,” Pacific Journal of Mathematics, vol. 157, no. 2, pp. 201–224, 1993. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  10. V. L. Makarov, M. J. Levin, and A. M. Rubinov, Mathematical Economic Theory: Pure and Mixed Types of Economic Mechanisms, vol. 33 of Advanced Textbooks in Economics, North-Holland, Amsterdam, The Netherlands, 1995. View at: Zentralblatt MATH | MathSciNet
  11. L. W. Mckenzie, “Turnpike theory,” Econometrica, vol. 44, no. 5, pp. 841–865, 1976. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. S. Pehlivan and M. A. Mamedov, “Statistical cluster points and turnpike,” Optimization, vol. 48, no. 1, pp. 93–106, 2000. View at: Publisher Site | Google Scholar | MathSciNet
  13. J. Connor, M. Ganichev, and V. Kadets, “A characterization of Banach spaces with separable duals via weak statistical convergence,” Journal of Mathematical Analysis and Applications, vol. 244, no. 1, pp. 251–261, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, John Wiley & Sons, New York, NY, USA, 4th edition, 1980. View at: Zentralblatt MATH | MathSciNet
  15. M. Burgin and O. Duman, “Statistical convergence and convergence in statistics,” preprint. View at: Google Scholar
  16. J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  17. J. A. Fridy, “Statistical limit points,” Proceedings of the American Mathematical Society, vol. 118, no. 4, pp. 1187–1192, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  18. J. A. Fridy and C. Orhan, “Statistical limit superior and limit inferior,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3625–3631, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  19. T. Šalát, “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, no. 2, pp. 139–150, 1980. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  20. E. Kolk, “The statistical convergence in Banach spaces,” Acta et Commentationes Universitatis Tartuensis, no. 928, pp. 41–52, 1991. View at: Google Scholar | MathSciNet
  21. S. Pehlivan and M. T. Karaev, “Some results related with statistical convergence and Berezin symbols,” Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 333–340, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  22. B. C. Tripathy, “On statistically convergent and statistically bounded sequences,” Malaysian Mathematical Society. Bulletin. Second Series, vol. 20, no. 1, pp. 31–33, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  23. B. C. Tripathy, “On statistically convergent sequences,” Bulletin of the Calcutta Mathematical Society, vol. 90, no. 4, pp. 259–262, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  24. G. Bachman and L. Narici, Functional Analysis, Academic Press, New York, NY, USA, 1966. View at: Zentralblatt MATH | MathSciNet

Copyright © 2007 Vinod K. Bhardwaj and Indu Bala. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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