Tauberian conditions for $L^{1}$-convergence of Fourier series

Abstract:

It is proved that Fourier series with asymptotically even coefficients and satisfying ${\lim _{\lambda \to 1}}\lim {\sup _{n \to \infty }}\sum _{j = n}^{[\lambda n]}{j^{p - 1}}|\Delta \hat f(j){|^p} = 0$, for some $1 < p \leqslant 2$, converge in ${L^1}$-norm if and only if $||\hat f(n){E_n} + \hat f( - n){E_{ - n}}|| = o(1)$, where ${E_n}(t) = \sum _{k = 0}^n{e^{ikt}}$. Recent results of Stanojević [1], Bojanic and Stanojević [2], and Goldberg and Stanojević [3] are special cases of some corollaries to the main theorem.