In this paper, a general theorem concerning absolute matrix summability is established by applying the concepts of almost increasing and δ-quasi-monotone sequences.

In this paper, a known theorem dealing with | C, 1 |k summability methods has been generalized under weaker conditions for | C, α, β ; δ |k summability methods. Some new results have also been obtained. AMS subject classifications: 40D15, 40F05, 40G99

In this paper, we prove two theorems on ∣A∣k, 1 ≤ k ≤ 2, summability of orthogonal series. The first one gives a sufficient condition under which an orthogonal series is absolutely summable almost everywhere, and the second one, is a general theorem, which also gives a sufficient condition so that an orthogonal series is absolutely summable almost everywhere, but it involves a positive numerica...

A sequence (bn) of positive numbers is said to be δ -quasi-monotone, if bn → 0, bn > 0 ultimately and ∆bn ≥ −δn, where (δn) is a sequence of positive numbers (see [3]). Let (φn) be a sequence of complex numbers and let ∑an be a given infinite series with partial sums (sn). We denote by σα n and tα n the nth Cesàro means of order α , with α > −1, of the sequences (sn) and (nan), respectively, i.e.,

We obtain necessary and (different) sufficient conditions for a series summable | ¯ N, p n | k , 1 < k ≤ s < ∞, to imply that the series is summable |T | s , where (¯ N, p n) is a weighted mean matrix and T is a lower triangular matrix. As corollaries of this result, we obtain several inclusion theorems. Let a n be a given series with partial sums s n , (C, α) the Césaro matrix of order α. If σ...

A triangle is a lower triangular matrix with all principal diagonal entriesbeing nonzero. Given an almost increasing sequence {Xn} and a sequence {λn} satisfying certain conditions, we obtain sufficient conditions for the series ∑ anλn to be absolutely summable of order k ≥ 1 by a triangle T . As a corollary we obtain the corresponding result when T is a weighted mean matrix. Theorem 1 of this ...