Convergence of subsequences of partial cubic sums of Fourier series in mean and almost everywhere

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.