On Riemann summability of functions

Abstract

1. Let k be a positive integer. With the usual terminology, the serieswill be said to be Riemann summable (R, k) to l ifconverges 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) bywhere λ = {λn} is a sequence of non-negative numbers increasing to ∞.