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. 
@article{MZM_2002_71_6_a1,
     author = {A. S. Belov},
     title = {Remarks on {Mean} {Convergence} {(Boundedness)} of {Partial} {Sums} of {Trigonometric} {Series}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {807--817},
     publisher = {mathdoc},
     volume = {71},
     number = {6},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a1/}
}
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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/