On the best approximation of some classes of~periodic~functions~in~the space $L_{2}$

We consider the set $L_{2}^{(r)}$ of $2\pi$-periodic functions $f\in L_{2}$ whose $(r-1)$-th order derivative is absolutely continuous, and the $r$-th order derivative $f^{(r)}\in L_{2}.$ We solve the extremal problem of finding an exact Jackson--Stechkin type constant that connects the best polynomial approximation of functions from $L_{2}^{(r)}$ with the average value of the generalized $m$-th order modulus of continuity of their derivative $f^{(r)}$ in the space $L_{2}.$ We also consider the classes $W_{m}^{(r)}(u)$ and $W_{m}^{(r)}(u,\Phi)$ of functions from $L_{2}^{(r)}$ such that the average value of the generalized $m$-th order modulus of continuity of their derivative $f^{(r)}$ is bounded from above by unity and, accordingly, by the value of some function $\Phi(u).$ We calculate the exact values of the known $n$-widths (according to Bernstein, to Gelfand, to Kolmogorov, linear, and projection) of the class $W_{m}^{(r)}(u).$ Then we solve the extremal problem of finding the exact value of the best approximation for the class $W_{m}^{(r)}(u,\Phi).$  The obtained results develop and complement some known results on the best approximation of various classes of functions in $L_{2}.$ In the paper, we use methods for solving extremal problems in normed spaces, as well as the method developed by V.\,M.~Tikhomirov
for estimating from below the $n$-widths of functional classes in Banach spaces.