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.