On Riemann summability of functions

1. Let k be a positive integer. With the usual terminology, the series will be said to be Riemann summable (R, k) to l if converges for all sufficiently small x, and tends to l as x → 0. Here we take (sin nx)/(nx) as meaning 1 when n = 0. A more general summability method which has been considered by various authors ((1), (2), (5), (6), (8), (9), (10)), and is usually denoted by (ℜ, λ,k) is obtained by replacing (2) by where λ = {λn} is a sequence of non-negative numbers increasing to ∞.