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Summability of Double Sequences and Double Series Over Non-Archimedean Fields: A Survey

  • P. N. Natarajan
  • Hemen Dutta
Chapter

Abstract

In this chapter, K denotes a complete, non-trivially valued, non-Archimedean field. We introduce a new definition of convergence of a double sequence and a double series (Natarajan and Srinivasan, Ann Math Blaise Pascal 9:85–100, 2002), which seems to be most suitable in the non-Archimedean context. We study some of its properties. We then present a very brief survey of the results, proved so far, pertaining to the Nörlund, weighted mean, and (M, λm,n) (or Natarajan) methods of summability for double sequences. In this chapter, a Tauberian theorem for the Nörlund method for double series is presented.

Keywords

Non-Archimedean field Double sequence Double series 4-Dimensional infinite matrix Conservative matrix Regular matrix Pringsheim Silverman–Toeplitz theorem Schur’s theorem Steinhaus theorem Nörlund method Weighted mean method (M, λm,n) (or Natarajan) method Tauberian theorem 

References

  1. 1.
    M. Zeltser, On conservative matrix methods for double sequence spaces. Acta Math. Hungar. 95, 225–242 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    B.V. Limaye, M. Zeltser, On the Pringsheim convergence of double series. Proc. Est. Acad. Sci. Phys. Math. 58(2), 108–121 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    M. Mursaleen, S.A. Mohiuddine, Convergence Methods for Double Sequences and Applications (Springer, Berlin, 2014)CrossRefGoogle Scholar
  4. 4.
    A. Aasma, H. Dutta, P.N. Natarajan, An Introductory Course in Summability Theory (Wiley, New York, 2017)CrossRefGoogle Scholar
  5. 5.
    P.N. Natarajan, V. Srinivasan, Silverman–Toeplitz theorem for double sequences and series and its application to Nörlund means in non-Archimedean fields. Ann. Math. Blaise Pascal 9, 85–100 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    G. Bachman, Introduction to p-Adic Numbers and Valuation Theory (Academic Press, London, 1964)Google Scholar
  7. 7.
    P.N. Natarajan, The Schur and Steinhaus theorems for 4-dimensional matrices in ultrametric fields. Comment. Math. Prace Mat. 51, 203–209 (2011)MathSciNetzbMATHGoogle Scholar
  8. 8.
    P.N. Natarajan, Some characterizations of Schur matrices in ultrametric fields. Comment. Math. Prace Mat. 52, 137–142 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    P.N. Natarajan, An addendum to the paper “Some characterizations of Schur matrices in ultrametric fields. Comment. Math. Prace Mat. 53, 81–82 (2013)zbMATHGoogle Scholar
  10. 10.
    P.N. Natarajan, The Steinhaus theorem for Toeplitz matrices in non-Archimedean fields. Comment. Math. Prace Mat. 20, 417–422 (1978)MathSciNetzbMATHGoogle Scholar
  11. 11.
    P.N. Natarajan, An Introduction to Ultrametric Summability Theory, 2nd edn. (Springer, Berlin, 2015)CrossRefGoogle Scholar
  12. 12.
    P.N. Natarajan, Some Tauberian Theorems in Non-Archimedean Fields, p-Adic Functional Analysis. Lecture Notes in Pure and Applied Mathematics, vol. 192 (Marcel Dekker, New York, 1997), pp. 297–303Google Scholar
  13. 13.
    P.N. Natarajan, A Tauberian theorem for the Nörlund means for double series in ultrametric fields. Indian J. Math. Proc. Sixth Dr. George Bachman Memorial Conf. 57, 33–38 (2015)Google Scholar
  14. 14.
    P.N. Natarajan, S. Sakthivel, Multiplication of double series and convolution of double infinite matrices in non-Archimedean fields. Indian J. Math. 50, 125–133 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    P.N. Natarajan, S. Sakthivel, Weighted means for double sequences in non-Archimedean fields. Indian J. Math. 48, 201–220 (2006)MathSciNetzbMATHGoogle Scholar
  16. 16.
    P.N. Natarajan, Product theorems for certain summability methods in non-Archimedean fields. Ann. Math. Blaise Pascal 10, 261–267 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    V.K. Srinivasan, On certain summation processes in the p-adic field. Indag. Math. 27, 319–325 (1965)MathSciNetCrossRefGoogle Scholar
  18. 18.
    P.N. Natarajan, Natarajan summability method for double sequences and double series. Adv. Dev. Math. Sci. 6, 9–17 (2014)CrossRefGoogle Scholar
  19. 19.
    P.N. Natarajan, Some properties of the Natarajan method of summability for double sequences in ultrametric fields. Int. J. Phys. Math. Sci. 6, 22–27 (2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • P. N. Natarajan
    • 1
    • 2
  • Hemen Dutta
  1. 1.Independent Research ProfessionalChennaiIndia
  2. 2.Department of MathematicsGauhati UniversityGuwahatiIndia

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