The best $m$-term trigonometric approximations of the classes of periodic multivariate functions of mixed smoothness

Exact in order estimates are obtained for the best $m$-term trigonometric approximations of the classes of periodic multivariate functions of mixed smoothness (Nikol'skii-Besov $B^{\boldsymbol{r}}_{p, \theta}$ and Sobolev $W^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}$ classes) in the space $B_{q,1}$, $2\leq p<q<\infty$. The norm in this space is not weaker than the $L_q$-norm. The obtained estimates in the multivariate case, in contrast to the univariate, differ in order from the respective estimates in the $L_q$-space.