Tauberian L1 -convergence classes of fourier series. I
It is shown that the Stanojević  necessary and sufficient conditions for L1-convergence of Fourier series of(FORMULA PRESENTED) 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 (FORMULA PRESENTED). For Fourier series with coefficients asymptotically even with respect to a sequence (FORMULA PRESENTED), and satisfying (FORMULA PRESENTED) necessary and sufficient conditions for L1-convergence are obtained. In particular for (FORMULA PRESENTED), an important corollary is obtained which connects smoothness of ⨍ with smoothness of (FORMULA PRESENTED). © 1983 American Mathematical Society.
L-convergence of fourier series
Bray, William O., and Časlav V. Stanojević. "Tauberian ��¹-convergence classes of Fourier series. I." Transactions of the American Mathematical Society 275, no. 1 (1983): 59-69.
Transactions of the American Mathematical Society