Geodesic
Absolute convergence of Fourier series with respect to bounded systems
Matematičeskie zametki, Tome 15 (1974) no. 3, pp. 363-370.

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In this paper we prove that for the piecewise-linear unit jump approximation the sum of the moduli of the Fourier coefficients with respect to an arbitrary complete orthonormal system, which is totally bounded, has, when averaged over sections, a lower bound of order $\log N$, where $N^{-1}$ is the approximation step. 
@article{MZM_1974_15_3_a1,
     author = {S. V. Bochkarev},
     title = {Absolute convergence of {Fourier} series with respect to bounded systems},
     journal = {Matemati\v{c}eskie zametki},
     pages = {363--370},
     publisher = {mathdoc},
     volume = {15},
     number = {3},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_15_3_a1/}
}
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S. V. Bochkarev. Absolute convergence of Fourier series with respect to bounded systems. Matematičeskie zametki, Tome 15 (1974) no. 3, pp. 363-370. http://geodesic.mathdoc.fr/item/MZM_1974_15_3_a1/