Summability of double sequences and double series over non-archimedean fields: A survey

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.