Tauberian $L^{1}$-convergence classes of Fourier series. I

It is shown that the Stanojević [2] necessary and sufficient conditions for ${L^1}$-convergence of Fourier series of $f \in {L^1}(T)$ can be reduced to the classical form. A number of corollaries of a recent Tauberian theorem are obtained for the subclasses of the class of Fourier coefficients satisfying ${n^\alpha }|\Delta \hat {f}(n)| = o(l) (n \to \infty )$ for some $0 < \alpha \leqslant \frac {1}{2}$. For Fourier series with coefficients asymptotically even with respect to a sequence $\{{l_n}\} ,{l_n} = o(n) (n \to \infty )$, and satisfying \[ l_n^{ - 1/q}{\left ({\sum \limits _{k = n}^{n + [n/{l_n}]} {{k^{p - 1}}|\Delta \hat f(k)} {|^p}} \right )^{1/p}} = o(1) \quad (n \to \infty ), \quad 1/p + 1/q = 1,\] necessary and sufficient conditions for ${L^1}$-convergence are obtained. In particular for ${l_n} = [\parallel {\sigma _n}(f) - f{\parallel ^{ - 1}}]$, an important corollary is obtained which connects smoothness of $f$ with smoothness of $\{\hat f(n)\}$.