Extension of functions in non-isotropic Nikolskii–Besov spaces and approximation of their derivatives

We consider non-isotropic Nikolskii and Besov spaces with norms defined using `$L_p$-averaged' moduli of continuity of functions of appropriate orders along the coordinate directions, instead of moduli of continuity of given orders for derivatives along these directions. We construct continuous linear maps from such spaces of functions defined in domains of certain type to the ordinary non-isotropic Nikolskii and Besov spaces on $ \mathbb{R}^d$ in such a way that these maps are function extension operators. Hence both kinds of spaces coincide on such domains. We also find the weak asymptotics of approximation characteristics related to the problem of recovering the derivative from the values of a function at a given number of points, Stechkin's problem for the differentiation operator, and the problem of width asymptotics for non-isotropic Nikolskii and Besov classes in these domains.