Sharp Inequalities between the Best Root-Mean-Square Approximations of Analytic Functions in the Disk and Some Smoothness Characteristics in the Bergman Space
Abstract:
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.
Keywords:sharp inequalities, best joint approximation, smoothness characteristics, exact constants, $n$-widths.
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