Research Article | Open Access
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
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 spaces.
- A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, UK, 1979.
- H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloquium Mathematicum, vol. 2, pp. 73–74, 1951.
- H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951.
- I. J. Schoenberg, “The integrability of certain functions and related summability methods,” The American Mathematical Monthly, vol. 66, no. 5, pp. 361–375, 1959.
- 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.
- 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.
- A. R. Freedman and J. J. Sember, “Densities and summability,” Pacific Journal of Mathematics, vol. 95, no. 2, pp. 293–305, 1981.
- 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.
- 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.
- 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.
- L. W. Mckenzie, “Turnpike theory,” Econometrica, vol. 44, no. 5, pp. 841–865, 1976.
- S. Pehlivan and M. A. Mamedov, “Statistical cluster points and turnpike,” Optimization, vol. 48, no. 1, pp. 93–106, 2000.
- 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.
- I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, John Wiley & Sons, New York, NY, USA, 4th edition, 1980.
- M. Burgin and O. Duman, “Statistical convergence and convergence in statistics,” preprint.
- J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985.
- J. A. Fridy, “Statistical limit points,” Proceedings of the American Mathematical Society, vol. 118, no. 4, pp. 1187–1192, 1993.
- 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.
- T. Šalát, “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, no. 2, pp. 139–150, 1980.
- E. Kolk, “The statistical convergence in Banach spaces,” Acta et Commentationes Universitatis Tartuensis, no. 928, pp. 41–52, 1991.
- 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.
- B. C. Tripathy, “On statistically convergent and statistically bounded sequences,” Malaysian Mathematical Society. Bulletin. Second Series, vol. 20, no. 1, pp. 31–33, 1997.
- B. C. Tripathy, “On statistically convergent sequences,” Bulletin of the Calcutta Mathematical Society, vol. 90, no. 4, pp. 259–262, 1998.
- G. Bachman and L. Narici, Functional Analysis, Academic Press, New York, NY, USA, 1966.
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.