Geodesic
Tauberian conditions for $L^1$-convergence of modified complex trigonometric sums
Lobachevskii journal of mathematics, Tome 13 (2003), pp. 39-44.

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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. 
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K. Kaur. Tauberian conditions for $L^1$-convergence of modified complex trigonometric sums. Lobachevskii journal of mathematics, Tome 13 (2003), pp. 39-44. http://geodesic.mathdoc.fr/item/LJM_2003_13_a3/

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