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The article examines nonisotropic Nikolskii and Besov spaces with normsdefined using $L_p$-averaged moduli of continuity of functions of appropriateorders along the coordinate directions, instead of moduli of continuity ofknown orders for derivative functions along the same directions. The authorbuilds continuous linear mappings of such spaces of functions defined indomains of certain type to ordinary nonisotropic Nikolskii and Besov spaces in$\mathbb{R}^d $ that are function extension operators, thus incurringcoincidence of both kinds of spaces in the said domains. The article alsoprovides weak asymptotics of approximation characteristics related to theproblem of derivative reconstruction from function values at a given number ofpoints, the S.B.Stechkin's problem for differential operator, and the problemof width asymptotics for nonisotropic Nikolskii-Besov classes in those domains.  The supplement provides a broad-sense definition of $\alpha$-type domain. Itis demonstrated that all of the earlier statements established by the authorwith respect to the tasks of function extension, derivative reconstruction fromfunction values, differential operator approximation with bounded operators,and width asymptotics for nonisotropic Nikolskii-Besov function classes definedin $\alpha$-type domains (in a narrow sense), remain valid in the sameformulation even for an $\alpha$-type domain as construed in a broad sense.

Extending functions from nonisotropic Nikolskii-Besov spaces and approximating their derivatives (with supplement)