Conditions under which the Convergence of a Sequence or its Certain Subsequences Follows from the Summability by Deferred Weighted Means

Abstract

Let (uk) be a sequence of real or complex numbers. First, we consider a real sequence (uk) and formulate one-sided Tauberian conditions, which are necessary and sufficient for the convergence of certain subsequences of (uk) to follow from its deferred weighted summability. These conditions are satisfied either if (uk) is deferred slowly decreasing or if (uk) obeys a Landau-type Tauberian condition. Second, we consider a complex sequence (uk) and present a two-sided Tauberian condition, which is necessary and sufficient in order that the convergence of certain subsequences of (uk) follow from its deferred weighted summability. This condition is satisfied either if (uk) is deferred slowly oscillating or if (uk) obeys a Hardy-type Tauberian condition. Finally, we extend these results to sequences in ordered linear spaces over the real numbers.