the object of this paper is to establish a summability factor theorem for summabilitya, , k  1 k  where a is the lower triangular matrix with non-negative entries satisfying certain conditions.

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...

In this paper, a main theorem dealing with | C, 1 |k summability factors has been generalized under more weaker conditions for | C,α, β |k summability factors. This theorem also includes some new results. Mathematics Subject Classification 2000: 40D15, 40F05, 40G05, 40G99.

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

A positive sequence (bn) is said to be almost increasing if there exists a positive increasing sequence cn and two positive constants A and B such that Acn ≤ bn ≤ Bcn (see [1]). Obviously every increasing sequence is almost increasing but the converse need not be true as can be seen from the example bn = ne(−1) n . Let ∑ an be a given infinite series with partial sums (sn). We denote by tn n-th...