On the approximation of periodic functions in $L_2$ and the values of the widths of certain classes of functions

The sharp Jackson–Stechkin inequalities are received, in which a special module of continuity $\widetilde{\Omega}_{m}(f; t)$ determined by Steklov's function is used instead the usual modulus of continuity of $m$th order $\omega_{m}(f; t)$. Such generalized modulus of continuity of $m$th order were introduced by V. A. Abilov and F. V. Abilova. The introduced modulus of continuity found their application in the theory of polynomial approximation in Hilbert space in the works by M. Sh. Shabozov and G. A. Yusupov, S. B. Vakarchuk and V. I. Zabutnaya and others.
While continuing and developing these direction for some classes of functions defined by modulus of continuity, the new values of $n$-widths in the Hilbert space $L_{2}$ were found.