On the Behaviour of a Series Associated with the Conjugate Series of a Fourier Series

Abstract

1.1. Definition. Let λ ≡ λ(ω) be continuous, differentiable, and monotonic increasing in (0, ∞) and let it tend to infinity as ω → ∞. Suppose that ∑ an (we write ∑ for throughout the present paper) is a given infinite series and letThe series ∑ an is said to be summable |R, λ, r|, where r > 0, ifwhere A is a fixed positive number (6, Definition B). Now, for r > 0, m < ω < m + 1,Hence, ∑ an is summable |R, λ, r|, where r > 0, if