On the relation between Lebesgue summability and some other summation methods

It is shown that if ∑ n = 1 N n | c n | = O ( N ) , then Lebesgue summability, ( C , β ) summability ( β > 0 ), Abel summability, Riemann summability, and ( γ , κ ) summability ( κ ⩾ 1 ) of the series ∑ n = 0 ∞ c n are all equivalent to one another.