Sharp inequalities between the best root-mean-square approximations of analytic functions in the disk and some smoothness characteristics in the Bergman space. (English. Russian original) Zbl 1477.41006

Math. Notes 110, No. 2, 248-260 (2021); translation from Mat. Zametki 110, No. 2, 266-281 (2021).
Summary: 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.


41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
30E10 Approximation in the complex plane
41A25 Rate of convergence, degree of approximation
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