  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 $L_q$.
