The Exact Inequalities of Jackson–Stechkin Type and the Width Values for Some Classes of Functions in $L_{2}$ Space

In this paper, some exact inequalities between the best approximations of periodic differentiable functions with trigonometric polynomials and generalized moduli of the continuity $\Omega_{m}$ of $m$-th order in $L_{2}[0,2\pi]$ space are found. Similar averaged characteristics of function smoothness in studying the important problems in the constructive theory of functions were considered by K. V. Runovskiy, E. A. Strogenko, V. G. Krotov, P. Osvald and many others. For some classes of functions defined by indicated moduli of continuity where the $r$-th derivatives are bounded by functions which satisfy certain constraints were obtained the exact values of Bernstein, Gelfand, Kolmogorov, linear and projection $n$-widths. Here is given an example of a majorant for which all the stated claims are fulfilled.