A class of $L^{1}$-convergence

Abstract:

It is proved that if the Fourier coefficients $\{ {a_n}\}$ of $f \in {L^1}(0, \pi )$ satisfy $({\ast }){n^{ - 1}}\sum \nolimits _{k = n}^{2n} {{k^p}|\Delta {a_n}|p = o(1)}$, for some $1 < p \leqslant 2$, then $||{s_n} - f|| = o(1)$, if and only if ${a_n}\lg n = o(1)$. For cosine trigonometric series with coefficients of bounded variation and satisfying $({\ast })$ it is proved that a necessary and sufficient condition for the series to be a Fourier series is $\{ {a_n}\} \in \mathcal {C}$, where $\mathcal {C}$ is the Garrett-Stanojević [4] class.