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STATISTICAL A-SUMMABILITY OF DOUBLE SEQUENCES AND A KOROVKIN TYPE APPROXIMATION THEOREM

  • Belen, Cemal (Department of Mathematics Faculty of Science Cumhuriyet University) ;
  • Mursaleen, Mohammad (Department of Mathematics Aligarh Muslim University) ;
  • Yildirim, Mustafa (Mustafa Yildirim Department of Mathematics Faculty of Science Cumhuriyet University)
  • Received : 2011.04.23
  • Published : 2012.07.31

Abstract

In this paper, we define the notion of statistical A-summability for double sequences and find its relation with A-statistical convergence. We apply our new method of summability to prove a Korovkin-type approximation theorem for a function of two variables. Furthermore, through an example, it is shown that our theorem is stronger than classical and statistical cases.

Keywords

References

  1. J. Boos, Classical and Modern Methods in Summability, Oxford University Press, New York, 2000.
  2. K. Demirci and S. Karakus, Statistical A-summability of positive linear operators, Math. Comput. Modelling 53 (2011), no. 1-2, 189-195. https://doi.org/10.1016/j.mcm.2010.08.003
  3. F. Dirik and K. Demirci, Korovkin-type approximation theorem for functions of two variables in statistical sense, Turkish J. Math. 34 (2010), no. 1, 73-83.
  4. O. Duman, E. Erkus, and V. Gupta, Statistical rates on the multivariate approximation theory, Math. Comput. Modelling 44 (2006), no. 9-10, 763-770. https://doi.org/10.1016/j.mcm.2006.02.009
  5. O. Duman, M. K. Khan, and C. Orhan, A-statistical convergence of approximating operators, Math. Inequal. Appl. 6 (2003), no. 4, 689-699.
  6. O. H. H. Edely and M. Mursaleen, On statistical A-summability, Math. Comput. Modelling 49 (2009), no. 3-4, 672-680. https://doi.org/10.1016/j.mcm.2008.05.053
  7. H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244.
  8. A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002), no. 1, 129-138. https://doi.org/10.1216/rmjm/1030539612
  9. H. J. Hamilton, Transformations of multiple sequences, Duke Math. J. 2 (1936), no. 1, 29-60. https://doi.org/10.1215/S0012-7094-36-00204-1
  10. G. H. Hardy, Divergent Series, Oxford Univ. Press, London, 1949.
  11. P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publ. Co., Delhi, 1960.
  12. F. Moricz, Statistical convergence of multiple sequences, Arch. Math. (Basel) 81 (2003), no. 1, 82-89. https://doi.org/10.1007/s00013-003-0506-9
  13. M. Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003), no. 1, 223-231. https://doi.org/10.1016/j.jmaa.2003.08.004
  14. M. Mursaleen and O. H. H. Edely, Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl. 293 (2004), no. 2, 532-540. https://doi.org/10.1016/j.jmaa.2004.01.015
  15. R. F. Patterson and E. Savas, Korovkin and Weierstrass approximation via lacunary statistical sequences, J. Math. Stat. 1 (2005), no. 2, 165-167. https://doi.org/10.3844/jmssp.2005.165.167
  16. A. Pringsheim, Zur theorie der zweifach unendlichen zahlenfolgen, Math. Ann. 53 (1900), no. 3, 289-321. https://doi.org/10.1007/BF01448977
  17. G. M. Robison, Divergent double sequences and series, Trans. Amer. Math. Soc. 28 (1926), no. 1, 50-73. https://doi.org/10.1090/S0002-9947-1926-1501332-5
  18. D. D. Stancu, A method for obtaining polynomials of Bernstein type of two variables, Amer. Math. Monthly 70 (1963), no. 3, 260-264. https://doi.org/10.2307/2313121
  19. V. I. Volkov, On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, Dokl. Akad. Nauk SSSR (N.S.) 115 (1957), 17-19.

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