Geodesic
Boundedness of the series of absolute values of blocks of sine series
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and equations of mathematical physics, Tome 283 (2013), pp. 252-256

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The series of absolute values of blocks of a sine series whose coefficients satisfy a certain weakened monotonicity condition are considered. It is shown that to ensure the bounded convergence of such a series, one can take the same blocks as for the Fourier series of functions of bounded variation. 
@article{TM_2013_283_a15,
     author = {S. A. Telyakovskii},
     title = {Boundedness of the series of absolute values of blocks of sine series},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {252--256},
     publisher = {mathdoc},
     volume = {283},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_283_a15/}
}
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S. A. Telyakovskii. Boundedness of the series of absolute values of blocks of sine series. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and equations of mathematical physics, Tome 283 (2013), pp. 252-256. http://geodesic.mathdoc.fr/item/TM_2013_283_a15/