### Integrability theorems for trigonometric series

We show that, if the coefficients (an) in a series ${a}_{0}/2+{\sum}_{n=1}^{\infty}{a}_{n}cos\left(nt\right)$ tend to 0 as n → ∞ and satisfy the regularity condition that ${\sum}_{m=0}^{\infty}{{\sum}_{j=1}^{\infty}[{\sum}_{n=j{2}^{m}}^{(j+1){2}^{m}-1}|{a}_{n}-{a}_{n+1}\left|\right]\xb2}^{1/2}<\infty $, then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series ${\sum}_{n=1}^{\infty}{b}_{n}sin\left(nt\right)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if ${\sum}_{n=1}^{\infty}\left|{b}_{n}\right|/n<\infty $. These conclusions were previously known to hold under stronger restrictions on the sizes of the differences...