A class of trigonometric series

Trigonometric series with coefficientsa _k → 0 under the condition $$(\exists p \in R,p > 1):\left( {\sum\nolimits_{n = 1}^\infty {\left\{ {\sum\nolimits_{k = n}^\infty {|\Delta a_k |p_{/n} } } \right\}^{1/p}< \infty } } \right)$$ are considered. It is shown that, under these conditions, the cosine series is a Fourier series for which the conditiona _n In n → 0 is the criterion for convergence in the metric of L. For the sine series, this is true under the further assumption that ∑ _n=1 ^∞ |a _n |/n<∞.