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

  • P. N. Natarajan
  • Hemen Dutta


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.


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 


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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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