Geodesic
The best approximation of periodic functions by trigonometric polynomials in $L^2$
Matematičeskie zametki, Tome 2 (1967) no. 5, pp. 513-522 
N. I. Chernykh. The best approximation of periodic functions by trigonometric polynomials in $L^2$. Matematičeskie zametki, Tome 2 (1967) no. 5, pp. 513-522. http://geodesic.mathdoc.fr/item/MZM_1967_2_5_a8/
@article{MZM_1967_2_5_a8,
     author = {N. I. Chernykh},
     title = {The best approximation of periodic functions by trigonometric polynomials in~$L^2$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {513--522},
     year = {1967},
     volume = {2},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1967_2_5_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Estimates are gotten for the best approximations in $L_2(0,2\pi)$ of a periodic function by trigonometric polynomials in terms of its $m$-th continuity modulus or in terms of the continuity modulus of its $r$-th derivative. The inequality $$ E_{n-1}(f)_{L_2}<(C_{2m}^m)^{-1/2}\omega_m(2\pi/n;f)_{L_2} \qquad (f\ne\mathrm{const}) $$ is proved, where the constant $(C_{2m}^m)^{-1/2}$ is unimprovable for the whole space $L_2(0,2\pi)$. Two titles are cited in the bibliography.