Geodesic
Trigonometric series with monotone coefficients
Matematičeskie zametki, Tome 22 (1977) no. 1, pp. 77-83

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Let $\{a_n\}$ be a monotonically decreasing sequence. Then each sequence $\{b_n\}$ such that $b_n\downarrow0$, $b_n\le a_n$, $n=1,2,\dots$, is a sequence of Fourier-Lebesgue coefficients with respect to the system $\{\cos nx\}$ if and only if the sequence $\sum_{n=1}^\infty\frac{a_n}n$ converges. 
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     author = {L. A. Balashov},
     title = {Trigonometric series with monotone coefficients},
     journal = {Matemati\v{c}eskie zametki},
     pages = {77--83},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_1_a8/}
}
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L. A. Balashov. Trigonometric series with monotone coefficients. Matematičeskie zametki, Tome 22 (1977) no. 1, pp. 77-83. http://geodesic.mathdoc.fr/item/MZM_1977_22_1_a8/