Geodesic
On the boundedness below of trigonometric polynomials of best approximation
Matematičeskie zametki, Tome 79 (2006) no. 6, pp. 870-878 Cet article a éte moissonné depuis la source Math-Net.Ru

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As A. S. Belov proved, the partial sums of an even $2\pi$-periodic function f expanded in a Fourier series with convex coefficients $\{a_n\}_{n=0}^\infty$, are uniformly bounded below if the conditions $a_n = O(n^{-1})$, $n\to\infty$, are satisfied; moreover, this assertion is no longer valid if the exponent $-1$ in this condition is replaced by a greater one. In this paper, we obtain analogs of these results for trigonometric polynomials of best approximation to the function $f$ in the metric of $L_{2\pi}^1$. 
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V. S. Kolesnikov. On the boundedness below of trigonometric polynomials of best approximation. Matematičeskie zametki, Tome 79 (2006) no. 6, pp. 870-878. http://geodesic.mathdoc.fr/item/MZM_2006_79_6_a4/

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[2] Belov A. S., “O chastnykh summakh trigonometricheskogo ryada s vypuklymi koeffitsientami”, Matem. zametki, 50:4 (1991), 21–27

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