Sharp Inequalities between the Best Root-Mean-Square Approximations of Analytic Functions in the Disk and Some Smoothness Characteristics in the Bergman Space

In Jackson–Stechkin type inequalities for the smoothness characteristic $\Lambda_m(f)$, $m\in\mathbb N$, we find exact constants determined by averaging the norms of finite differences of $m$th order of a function $f\in B_2$. We solve the problem of best joint approximation for a certain class of functions from $B_2^{(r)}$, $r\in\mathbb Z_+$ whose smoothness characteristic $\Lambda_m(f)$ averaged with a given weight is bounded above by the majorant $\Phi$. The exact values of $n$-widths of some classes of functions are also calculated.