Trigonometric series with monotone coefficients

Let $\{a_n\}$ be a monotonically decreasing sequence. Then each sequence $\{b_n\}$ such that $b_n\downarrow0$, $b_n\le a_n$, $n=1,2,\dots$, is a sequence of Fourier-Lebesgue coefficients with respect to the system $\{\cos nx\}$ if and only if the sequence $\sum_{n=1}^\infty\frac{a_n}n$ converges.