Geodesic
Non-Fourier-Lebesgue trigonometric series with nonnegative partial sums
Matematičeskie zametki, Tome 59 (1996) no. 1, pp. 24-41

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It is proved that a trigonometric cosine series of the form $\sum_{n=0}^\infty a_n\cos(nx)$ with nonnegative coefficients can be constructed in such a way that all of its partial sums are positive on the real axis. It converges to zero almost everywhere and is not a Fourier-Lebesgue series. Some other properties of trigonometric series with nonnegative partial sums are also studied. 
@article{MZM_1996_59_1_a2,
     author = {A. S. Belov},
     title = {Non-Fourier-Lebesgue trigonometric series with nonnegative partial sums},
     journal = {Matemati\v{c}eskie zametki},
     pages = {24--41},
     publisher = {mathdoc},
     volume = {59},
     number = {1},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_1_a2/}
}
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A. S. Belov. Non-Fourier-Lebesgue trigonometric series with nonnegative partial sums. Matematičeskie zametki, Tome 59 (1996) no. 1, pp. 24-41. http://geodesic.mathdoc.fr/item/MZM_1996_59_1_a2/