On Estimates in~$L_2(\mathbb{R})$ of Mean $\nu$-Widths of Classes of Functions Defined via the Generalized Modulus of Continuity of~$\omega_{\mathcal{M}}$

S. B. Vakarchuk

Matematičeskie zametki, Tome 106 (2019) no. 2, pp. 198-211.

Résumé

For the classes of functions $$ W^r(\omega_{\mathcal{M}},\Phi):=\{f \in L^r_2(\mathbb{R}): \omega_{\mathcal{M}}(f^{(r)},t) \le \Phi(t) \ \forall\,t \in (0,\infty)\}, $$ where $\Phi$ is a majorant and $r \in \mathbb{Z}_{+}$, lower and upper bounds for the Bernstein, Kolmogorov, and linear mean $\nu$-widths in the space $L_2(\mathbb{R})$ are obtained. A condition on the majorant $\Phi$ under which the exact values of these widths can be calculated is indicated. Several examples illustrating the results are given.

Détail

Keywords: mean dimension, mean $\nu$-width, entire function of exponential type, generalized modulus of continuity.
Mots-clés : majorant

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