A class of trigonometric series

Trigonometric series with coefficients $a_k\to0$ under the condition
$$ (\exists\,p\in R,p>1):\biggl(\sum_{n=1}^\infty\biggl\{\sum_{k=n}^\infty|\Delta a_k|^p/n\biggr\}^{1/p}<\infty\biggr). $$
are considered. It is shown that, under these conditions, the cosine series is a Fourier series for which the condition $a_n\ln n\to0$ is the criterion for convergence in the metric of $L$. For the sine series, this is true under the further assumption that $\sum_{n=1}^\infty|a_n|/n<\infty$.