Classes of $L^{1}$-convergence of Fourier and Fourier-Stieltjes series

It is shown that the Fomin class ${\mathcal {F}_p}(1 < p \leqslant 2)$ is a subclass of $\mathcal {C} \cap \mathcal {B}\mathcal {V}$, where $\mathcal {C}$ is the Garrett-Stanojević class and $\mathcal {B}\mathcal {V}$ the class of sequences of bounded variation. Wider classes of Fourier and Fourier-Stieltjes series are found for which ${a_n}\;{\text {lg}}\;n = o(1),n \to \infty$, is a necessary and sufficient condition for ${L^1}$-convergence. For cosine series with coefficients in $\mathcal {B}\mathcal {V}$ and $n\Delta {a_n} = O(1)$, $n \to \infty$, necessary and sufficient integrability conditions are obtained.