Geodesic
Convergence of subsequences of partial cubic sums of Fourier series in mean and almost everywhere
Matematičeskie zametki, Tome 52 (1992) no. 3, pp. 63-77.

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Under certain assumptions on the regularity of a function $\Phi$ necessary and sufficient conditions are found for $\Phi$ under which the integrability of $\Phi(|f|)$ implies, for every function $f$ measurable on $T^d$, the existence of a subsequence of cubic sums of the Fourier series of $f$ that converges to f in mean or almost everywhere. 
@article{MZM_1992_52_3_a6,
     author = {S. V. Konyagin},
     title = {Convergence of subsequences of partial cubic sums of {Fourier} series in mean and almost everywhere},
     journal = {Matemati\v{c}eskie zametki},
     pages = {63--77},
     publisher = {mathdoc},
     volume = {52},
     number = {3},
     year = {1992},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1992_52_3_a6/}
}
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S. V. Konyagin. Convergence of subsequences of partial cubic sums of Fourier series in mean and almost everywhere. Matematičeskie zametki, Tome 52 (1992) no. 3, pp. 63-77. http://geodesic.mathdoc.fr/item/MZM_1992_52_3_a6/