Tauberian $L\sp{1}$-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 ${n^\alpha }\vert\Delta \hat{f}(n)\vert = o(l)\,(n \to \infty)$ 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

$$l_n^{ - 1/q}{\left({\sum\limits_{k = n}^{n + [n/{l_n}]} {{k^{p - ... ...} {\vert^p}} \right)^{1/p}} = o(1)\, \quad (n \to \infty), \quad 1/p + 1/q = 1,$$

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 $\{\hat f(n)\}$.