Necessary and Sufficient Tauberian Conditions, Under Which Convergence Follows From Summability (C, 1)

Let (s_k: k = 0, 1, ...) be a sequence of real numbers whichis summable (C, 1) to a finite limit. We prove that (s_k) isconvergent if and only if the following two conditions are satisfied: where _n denotes the integer part of the productn. Both conditions are clearly satisfied if (s_k) is slowly decreasingin the sense of R. Schmidt and G. H. Hardy. The symmetric counterparts of the conditions above are thosewhen ‘limsup’ and ‘liminf’ are interchangedon the left-hand sides, while the inequality sign ‘ ’is changed for the opposite ‘ ’ in them. Next, let (s_k) be a sequence of complex numbers which is summable(C, 1) to a finite limit. We prove that (s_k) is convergent ifand only if one of the following conditions is satisfied: We also prove a general Tauberian theorem forsequences in ordered linear spaces.