Inequalities between Best Polynomial Approximants and Smoothness Characteristics of Functions in $L_2$

Exact constants in Jackson–Stechkin type inequalities are found for the smoothness characteristics $\Lambda_m (f)$, $m\in\mathbb N$, determined by averaging the norm of finite differences of $m$th order of functions $f \in L_2$. A solution is given of the extremal problem of finding the supremum for best joint polynomial approximations of functions and their successive derivatives on some classes of functions from $L_2$ whose averaged norms of finite differences are bounded above by $1$.