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

Source 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 ∑ a n (we write ∑ for throughout the present paper) is a given infinite series and let The series ∑ a n is said to be summable | R , λ, r |, where r > 0, if where A is a fixed positive number ( 6 , Definition B). Now, for r > 0, m < ω < m + 1, Hence, ∑ a n is summable | R , λ, r |, where r > 0, if