Jackson-type inequalities with generalized modulus of continuity and exact values of the $n$-widths for the classes of $(ψ,β)$-differentiable functions in $L_2$. III

Abstract

In the classes $L^{\psi}_{\beta ,2}$ of $2\pi$ -periodic $(\psi , \beta)$-differentiable functions for which $f^{\psi}_{\beta} \in L_2$, we determine the exact constants in Jackson-type inequalities for the characteristic of smoothness $\Lambda_{\gamma} (f, t) = \biggl\{\frac1t \int^t_0 \| \Delta^{\gamma}_ h(f)\|^2dh \biggr\}^{1/2}$, $t > 0$, deternined by averaging the norm of the generalized difference relation $\Delta_{ \gamma}h(f)$. For the classes of $(\psi,\beta)$ -differentiable functions defined by using the characteristic of smoothness $\Lambda_{\gamma}$ and the majorant $\Phi$, satisfying numerous conditions, we find the exact values of some $n$-widths in $L_2$.