Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in $L_2$

We find the exact values of the $n$-widths for the classes of periodic differentiable functions in $L_2[0,2\pi]$, satisfying the constraint
$$ \int_0^ht\widetilde\Omega_m^{1/m}(f^{(r)};t)\,dt\le\Phi(h), $$
where $h>0$, $m\in\mathbb N$, $r\in\mathbb Z_+$, $\widetilde\Omega_m^{1/m}(f^{(r)};t)$ is the generalized $m$th order continuity modulus of the derivative $f^{(r)}\in L_2[0,2\pi]$, while $\Phi(t)$ is an arbitrary increasing function such that $\Phi(0)=0$.