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Natarajan, P. N.; Dutta, Hemen

Summability of double sequences and double series over non-Archimedean fields: a survey. (English) Zbl 1450.40005

Dutta, Hemen (ed.) et al., Current trends in mathematical analysis and its interdisciplinary applications. Cham: Birkhäuser. 715-736 (2019).
Summary: 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 [the first author and V. Srinivasan, Ann. Math. Blaise Pascal 9, No. 1, 85–100 (2002; Zbl 1009.40002)], 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 \((M, \lambda_m,n)\) (or Natarajan) methods of summability for double sequences. In this chapter, a Tauberian theorem for the Nörlund method for double series is presented.
For the entire collection see [Zbl 1425.35003].

MSC:

40B05 Multiple sequences and series
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
40E05 Tauberian theorems
26E30 Non-Archimedean analysis
40-02 Research exposition (monographs, survey articles) pertaining to sequences, series, summability

Keywords:

non-Archimedean field; double sequence; double series; 4-dimensional infinite matrix; conservative matrix; regular matrix; Pringsheim convergence; Silverman-Toeplitz theorem; Schur’s theorem; Steinhaus theorem; Nörlund method; weighted mean method; \((M, \lambda_m, n)\) method; Natarajan method; Tauberian theorem

Citations:

Zbl 1009.40002
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Full Text: DOI

References:

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