Extension of Functions from Isotropic Nikol’skii–Besov Spaces and Their Approximation together with Derivatives

Abstract

Isotropic Nikol’skii–Besov spaces with norms whose definition, instead of the modulus of continuity of certain order of partial derivatives of functions of fixed order, involves the “$L_p$-averaged” modulus of continuity of functions of the corresponding order are studied. We construct continuous linear mappings of such spaces of functions given on bounded domains of $(1,\dots,1)$-type (in the wide sense) to the usual isotropic Nikol’skii–Besov spaces on $\mathbb{R}^d$, which are extension operators of these functions; this implies the coincidence of these spaces on the domains mentioned above. It is established that any bounded domain in $\mathbb{R}^d$ with Lipschitz boundary is a domain of $(1,\dots,1)$-type (in the wide sense). We also establish the weak asymptotics of approximation characteristics related to the problem of the recovery of functions together with their derivatives from the values of these functions at a given number of points, to the Stechkin problem for the differentiation operator, and to the problem of describing the asymptotics of widths for isotropic classes of Nikol’skii and Besov in these domains.