Jackson-Stechkin Inequality and Values of Widths of Some Classes of Functions in $L_2$

The sharp values of extremal characteristic of special form for classes $L_{2}^{(r)}$, $(r\in\mathbb{Z}_{+})$ containing not only averaged module of continuity but also the averaged with weight $u(t-u)/t$, $0\le u\le t$ of given modulus continuity is calculated. The obtained result is the spreading of well-known S. B. Vakarchuk theorem about averaged module of continuity. For the given characteristic of smoothness, is given an application for the solution of one extremal problem and the values of $n$-widths for some classes of functions in $L_2$ is calculated.