On generalized characteristics of smoothness of functions and on average $\nu$-widths in the space $L_2(\mathbb{R})$

Abstract

Estimates above and estimates below have been obtained for Kolmogorov, linear and Bernshtein average $\nu$-widths on the classes of functions $W^r (\omega^w, \Psi)$, where $r \in \mathbb{N}$, $\omega^w(f)$ is the generalized characteristic of smoothness of a function $f \in L_2(\mathbb{R})$, $\Psi$ is a majorant. Exact values of the enumerated extremal characteristics of approximation, following from the one condition on the majorant were obtained too.

Keywords

• generalized modulus of continuity
• majorant
• entire function
• average $\nu$-width