A Tauberian theorem for absolute summability

Abstract

Let be the given infinite series with {sn} as the sequence of partial sums and let be the binomial coefficient of zn in the power series expansion of the function (l-z)-σ-1 |z| < 1. Now let, for β > – 1,converge for 0 ≤ x < 1. If fβ(x) → s as x → 1–, then we say that ∑an is summable (Aβ) to s. If, further, f(x) is a function of bounded variation in (0, 1), then ∑an is summable |Aβ| or absolutely summable (Aβ). We write this symbolically as {sn} ∈ |Aβ|. This method was first introduced by Borwein in (l) where he proves that for α > β > -1, (Aα) ⊂ (Aβ). Note that for β = 0, (Aβ) is the same as Abel method (A). Borwein (2) also introduced the (C, α, β) method as follows: Let α + β ╪ −1, −2, … Then the (C, α, β) mean is defined by