The best polynomial approximation, derivatives of fractional order, and widths of classes of functions in $L_2$

On the classes of $2\pi$-periodic functions ${\mathcal{W}}^{\alpha} (K_{\beta}, \Phi)$, where $\alpha, \beta \in (0;\infty)$, defined by $K$-functionals $K_{\beta}$, fractional derivatives of order $\alpha$, and majorants $\Phi$, the exact values of different $n$-widths have been computed in the space $L_2$.