Tauberian conditions for $L^1$-convergence of modified complex trigonometric sums

An $L^1$-convergence property of the complex form $g_n(c,t)=S_n(c,t)-[c_nE_n(t)+c_{-n}E_{-n}(t)]$ of the modified sums introduced by Garrett and Stanojević [3] is established and a necessary and sufficient condition for $L^1$-convergence of Fourier series is obtained.