2019
Том 71
№ 6

# Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications

Abstract

We show that the well-known results on estimates of upper bounds of functionals on the classes $W^r H^{ω}$ of periodic functions can be regarded as a special case of Kolmogorov-type inequalities for support functions of convex sets. This enables us to prove numerous new statements concerning the approximation of the classes $W^r H^{ω}$, establish the equivalence of these statements, and obtain new exact inequalities of the Bernstein-Nikol’skii type that estimate the value of the support function of the class $H^{ω}$ on the derivatives of trigonometric polynomials or polynomial splines in terms of the $L^{ϱ}$ -norms of these polynomials and splines.

English version (Springer): Ukrainian Mathematical Journal 52 (2000), no. 1, pp January 2000, Volume.

Citation Example: Babenko V. F., Kofanov V. A., Korneichuk N. P., Pichugov S. A. Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications // Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 66-84.

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