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

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