Geodesic
Remarks on Mean Convergence (Boundedness) of Partial Sums of Trigonometric Series
Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 807-817.

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Fairly general conditions on the coefficients $\{a_n\}_{n=1}^\infty$ of even and odd trigonometric Fourier series under which $L$-convergence (boundedness) of partial sums of the series is equivalent to the relation $\sum _{k=[n/2]}^{2n}|a_k|/(|n-k|+1)=o(1)$ ($=O(1)$, respectively) are given. 
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A. S. Belov. Remarks on Mean Convergence (Boundedness) of Partial Sums of Trigonometric Series. Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 807-817. http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a1/

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