Exact Values of Widths of Some Functional Classes in $L_{2}$ and Minimization of the Constants in Inequalities of Jackson–Stechkin Type

In this paper, it is considered the extremal problem of finding the exact constants in inequalities of Jackson–Stechkin type between the best approximations of periodic differentiable functions $f\in L_{2}^{(r)}[0,2\pi]$ by trigonometric polynomials, and the average values with a positive weight $\varphi$ moduli of continuity of $m$th order $\omega_{m}(f^{(r)}, t),$ belonging to the space $L_{p},\, 0<p\le2$. In particular, the problem of minimizing the constants in these inequalities over all subspaces of dimension $n,$ raised by N. P. Korneychuk, is solved. For some classes of functions defined by the specified moduli of continuity, the exact values of $n$-widths of class
\begin{equation*} L_{2}^{(r)}(m,p,h;\varphi):=\left\{f\in L_{2}^{(r)}: \left(\int\limits_{0}^{h}\omega_{m}^{p}(f^{(r)};t)_{2}\,\varphi(t)dt\right)^{1/p} \hspace{-1.7mm}\left(\int\limits_{0}^{h}\varphi(t)dt\right)^{-1/p}\le1\right\} \end{equation*}
are found in the Hilbert space $L_2,$ and the extreme subspace is identified. In this article, the results are shown which are the extension and the generalization of some earlier results obtained in this line of investigation.