Concerning a sufficient condition of Sidon for the integrability of trigonometric series

We consider problems concerning integrability and convergence in the metric $L$ of trigonometric series, the coefficients $a_k$ of which satisfy the conditions: $a_k\to0$, there exist numbers $A_k$ such that $A_k\downarrow0$, $\sum A_k<\infty$, and $|\Delta a_k|\le A_k$. The integrability of a series in cosines under these conditions is equivalent to a theorem of Sidon.