Approximation characteristics of Nikol'skii-Besov and Sobolev classes in the space $B_{q,1}$

Abstract

We investigate Kolmogorov width and entropy numbers of the Nikol'skii--Besov classes $B^{\boldsymbol{r}}_{p,\theta}$ and Sobolev classes $W^{\boldsymbol{r}}_{p, \boldsymbol{\alpha}}$ of periodic functions of mixed smoothness of one and many variables in the space $B_{q,1}$. The norm in this space is ``stronger'' than the respective$L_q$-norm.The paper considers the case $2 \leq p < q < \infty$, which has remained unexplored so far, since it required a~significantly different method for getting upper estimates of the Kolmogorov width. It was shown that in this particular case of relations between the parameters $p$ and $q$,the exact-order estimates of the Kolmogorov widths of both of the considered classes in the space $B_{q,1}$ are not realized by an approximation by the subspace of trigonometric polynomials with ``numbers'' of harmonics from the step hyperbolic crosses of the corresponding dimension. Besides, in the multidimensional case $(d \geq 2)$, in contrast to the one-dimensional case, the orders of the respective Kolmogorov widths and entropy numbers in the space $B_{q,1}$ differ from the corresponding orders of these approximation characteristics for the $L_q$-space.