On Tauberian theorems for (A, k) summability method

Abstract

ABSTRACT. Let (u n ) be a sequence of real numbers. In this paper we introduce some Tauberian conditions in terms of regularly generated sequences for (A, k) summability method.

Some one-sided conditions under which subsequential convergence follows from (A,k) summability method

“…After taking j = k − 1 in (4) and r = m in Theorem 3.2, we obtain (S k−1 (I − S)u) n → 0, n → ∞. (6) Continuing in this vein, we get (S(I − S)u) n → 0, n → ∞. Since every Cesàro summable sequence is Abel summable, ((I − S)u) n is Abel summable to 0.…”

“…Our aim in this work is to generalize the results in [6] and recover subsequential convergence of sequences under some weaker conditions than those given in [6]. Moreover, convergence of a sequence will be recovered as a special case of the result in this work.…”

“…Recently, Çanak and Totur [2] have given two new identities for (2) and then shown that the condition in the main theorem of Stanojević and Stanojević [5] can be replaced by the condition that (2) be left one-sidedly bounded with respect to some nonnegative sequence. Çanak et al [6] showed that one can recover convergence of a sequence from its (A, k) summability under some more general conditions than those given in [7]. However, some conditions do not imply convergence of a sequence out of (A, k) summability.…”

Extended Tauberian conditions for (C,1) summability method

“…Instead of recovering convergence of sequence (u n ) out of the existence of (1.4) and some additional conditions imposed on the sequence (u n ), one can obtain further information about the asymptotic behavior of the sequence (u n ) by (C, 1) summability of its generator sequence. In light of this reasoning, Çanak [10] and Çanak and Dik [11] obtained slow oscillation of (u n ) out of the existence of (C, 1) summability of its generator sequence with additional suitable conditions on the sequence (u n ).…”

Convergence and subsequential convergence of regularly generated sequences

“…Recently, several results in terms of regularly generated sequences for different purposes have been obtained by Dik et al [10], Ç anak et al [1], Ç anak and Totur [3], Ç anak et al [2], Ç anak et al [7], Ç anak and Totur [5] and many more. In this paper, we first recover convergence and subsequential convergence of a sequence which is regularly generated by another sequence in some sequence spaces under certain conditions.…”