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

P. N. Natarajan () and Hemen Dutta ()
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P. N. Natarajan: Independent Research Professional

Chapter Chapter 18 in Current Trends in Mathematical Analysis and Its Interdisciplinary Applications, 2019, pp 715-736 from Springer

Abstract: 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, λ m,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; λ m; n) (or Natarajan) method; Tauberian theorem (search for similar items in EconPapers)
Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-15242-0_18

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DOI: 10.1007/978-3-030-15242-0_18

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