Inclusion and equivalence relations between absolute Norlund and absolute weighted mean summability methods

On Inclusion Relations for Absolute Summability

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...

Inclusion Theorems for Absolute Summability Methods

In this paper we have proved two theorems concerning an inclusion between two absolute summability methods by using any absolute summability factor.

On absolute generalized Norlund summability of double orthogonal series

In the paper [Y. Okuyama, {it On the absolute generalized N"{o}rlund summability of orthogonal series},Tamkang J. Math. Vol. 33, No. 2, (2002), 161-165] the author has found some sufficient conditions under which an orthogonal seriesis summable $|N,p,q|$ almost everywhere. These conditions are expressed in terms of coefficients of the series. It is the purpose ofthis paper to extend this result...

Necessary and sufficient conditions for inclusion relations for absolute summability

In this paper we obtain appropriate necessary and sufficient conditions for |N, pn|k summability to imply that of |N,qn|s for 1< k≤ s<∞. As in [6] we make use of a result of Bennett [1], who has obtained necessary and sufficient conditions for a factorable matrix to map lk → ls. A factorable matrix A is one in which each entry ank = bnck. Weighted mean matrices are factorable. It will not be po...

Absolute matrix summability methods