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

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 $B_{p,\theta }^{\mathbf{r}}$) in the space $B_{q,1}$, $1\le p,q\le \infty$, 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 $B_{p,\theta }^{\mathbf{r}}$ in the space $B_{q,1}$ comparing to the known estimates in the space $L_q$.