Geodesic
On Conditions of the Average Convergence (Upper Boundedness) of Trigonometric Series
Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 8-20.

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Let $c_n=\widehat f(n)$ be Fourier coefficients of a function $f\in L_{2\pi}$. We prove that the condition $$ \sum_{k=\left[\frac n2\right]}^{2n}\frac{|c_k|+|c_{-k}|}{|n-k|+1}=o(1) \quad \big(=O(1)\big) $$ is necessary for the convergence of the Fourier series of $f$ in the $L$-metric; moreover, this condition is sufficient under some additional hypothesis for Fourier coefficients of $f$. 
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A. S. Belov. On Conditions of the Average Convergence (Upper Boundedness) of Trigonometric Series. Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 8-20. http://geodesic.mathdoc.fr/item/CMFD_2007_25_a1/

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