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 Norlund, weighted mean, and (M, λm,n) (or Natarajan) methods of summability for double sequences. In this chapter, a Tauberian theorem for the Norlund method for double series is presented.