The exact Jackson–Stechkin inequality in $L_{2,\mu_{\alpha}}$

Several extremal problems on the best mean-square approximation of the functions $f,$ on a semiaxis with a power-law weight are solved in the paper, which can be applied in solving various problems. Exact Jackson–Stechkin-type inequalities are obtained for some classes of functions in which the values of the best approximations are estimated from above in terms of $k$-th order Hankel moduli of smoothness.