Reconstructing derivatives from values of functions belonging to Nikolskii-Besov classes of mixed smoothness in domains of a certain kind

TL;DR

This paper investigates the approximation of derivatives of functions in Nikolskii-Besov classes with mixed smoothness, providing new bounds on reconstruction accuracy in specific domains, and broadening the applicable function classes.

Contribution

It introduces new upper and lower bounds for derivative reconstruction accuracy, extending previous results to a wider class of Nikolskii-Besov spaces with mixed smoothness.

Findings

Derived bounds are sometimes stronger than previous estimates.

Extended the class of Nikolskii-Besov spaces with known reconstruction bounds.

Provided estimates for functions in bounded 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 of the best accuracy of reconstruction derivatives from function values at a given number of points for such classes of functions in bounded 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.