A Tauberian theorem for absolute summability

Abstract

Let be the given infinite series with { s n } as the sequence of partial sums and let be the binomial coefficient of z n 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 f β ( x ) → s as x → 1–, then we say that ∑ a n is summable ( A β ) to s . If, further, f ( x ) is a function of bounded variation in (0, 1), then ∑ a n is summable | A β | or absolutely summable ( Aβ ). We write this symbolically as { s n } ∈ |Aβ|. This method was first introduced by Borwein in (l) where he proves that for α > β > -1, ( A α ) ⊂ ( A β ). Note that for β = 0, ( A β ) 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