Estimates of the characteristics of nonlinear approximation of classes of periodic functions of many variables

In the paper, we obtained exact order estimates of the best $m$-term trigonometric approximation and the best orthogonal trigonometric approximation of functions from the Nikol'skii-Besov $B^{\boldsymbol{r}}_{p,\theta}(\mathbb{T}^d)$ and Sobolev $W^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}(\mathbb{T}^d)$ classes in the Lebesque subspaces $B_{q,1}(\mathbb{T}^d)$ for some relations between the parameters $p$ and $q$. The received results yield that estimates of the considered approximation characteristics in the multivariate case, in contrast to the univariate, in the spaces $B_{q,1}(\mathbb{T}^d)$ and $L_q(\mathbb{T}^d)$ differ in order. Besides, for some parameter values the obtained exact order estimates of the best $m$-term trigonometric approximation and the best orthogonal trigonometric approximation in the spaces $B_{q,1}(\mathbb{T}^d)$ still remain unknown for the $L_q(\mathbb{T}^d)$-space.