Estimates for the approximation characteristics of the Nikol'skii-Besov classes of functions with mixed smoothness in the space $B_{q,1}$

TL;DR

This paper derives precise estimates for approximation characteristics of multivariate Nikol'skii-Besov function classes with mixed smoothness in the stronger $B_{q,1}$ space, highlighting differences from $L_q$ and advancing existing bounds.

Contribution

It provides exact-order estimates for approximation characteristics of Nikol'skii-Besov classes in the $B_{q,1}$ space, improving upon previous $L_q$ estimates and emphasizing multivariate distinctions.

Findings

Approximate estimates differ in order from $L_q$ in multivariate cases.

Significant progress in approximation estimates for $B^{oldsymbol{r}}_{p, heta}$ classes in $B_{q,1}$.

Enhanced understanding of approximation in stronger function spaces.

Abstract

Exact-order estimates are obtained for some approximation characteristics of the classes of periodic multivariate functions with mixed smoothness (the Nikol'skii-Besov classes $B^{\boldsymbol{r}}_{p, \theta}$) in the space $B_{q,1}$, $1 \leq p, q \leq \infty$, which norm is stronger than the $L_q$-norm. It is shown, that in the multivariate case (in contrast to the univariate) in most of the considered situations the obtained estimates differ in order from the corresponding estimates in the space $L_q$. Besides, a significant progress is made in estimates for the considered approximation characteristics of the classes $B^{\boldsymbol{r}}_{p, \theta}$ in the space $B_{q, 1}$ comparing to the known estimates in the space $L_q$.