The best approximation of periodic functions by trigonometric polynomials in $L^2$

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.