Jackson-Type Inequalities and Widths of Function Classes in $L_2$

The sharp Jackson-type inequalities obtained by Taikov in the space $L_2$ and containing the best approximation and the modulus of continuity of first order are generalized to moduli of continuity of $k$th order $(k=2,3,\dots)$. We also obtain exact values of the $n$-widths of the function classes $F(k,r,\Phi)$ and $\mathcal{F}_k^r (h)$, which are a generalization of the classes $F(1,r,\Phi)$ and $\mathcal{F}^r_1(h)$ studied by Taikov.