An $L^{1}$--convergence property of the complex form $g_{n}(c,t)=S_{n}(c,t)- \left[ c_{n}E_{n}(t)+c_{-n}E_{-n}(t) \right] $ of the modified sums introduced by Garrett and Stanojevi\'{c} [3] is established and a necessary and sufficient\ condition for $L^{1}$-convergence of Fourier series is obtained.
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