Approximation of partial differential operators on Nikolskii-Besov functions classes of mixed smoothness in domains of a certain kind

Abstract

The article examines Nikolskii and Besov spaces with norms defined using "$L_p$-averaged" mixed moduli of continuity for functions of appropriate orders, instead of mixed moduli of continuity of known orders for certain mixed derivative functions. The work provides upper and lower estimates for a quantity of the best accuracy of approximation in the S.B.Stechkin problem for partial differential operators on a classes of such functions defined in domains of a certain kind. These estimates are not weaker, but in some cases even stronger than those derived by the author in the problem under consideration for the aforementioned classes of functions on cube $ I^d. $ It also significantly broadens the class of Nikolskii and Besov spaces of mixed smoothness for which mentioned estimates in the problem under consideration have been derived.