A~class of trigonometric series

G. A. Fomin

Matematičeskie zametki, Tome 23 (1978) no. 2, pp. 213-222.

Résumé

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 $\sum_{n=1}^\infty|a_n|/n\infty$.