Estimates of $M$–term approximations of functions of several variables in the Lorentz space by a constructive method

In the paper, the Lorentz space $L_{q,r}(\mathbb{T}^m)$ of periodic functions of several variables, the Nikol'skii–Besov class $S_{q,\tau,\theta}^{\overline{r}}$ and the associated class $W_{q,r}^{a,b,\overline{r}}$ for $1<q$, $\tau<\infty$, $1\leqslant\theta\leqslant\infty$ are considered. Estimates are established for the best $M$-term trigonometric approximations of functions of the classes $W_{q,\tau_1}^{a,b,\overline{r}}$ and $S_{q,\tau_1,\theta}^{\overline{r}}B$ in the norm of the space $L_{p,\tau_2}(\mathbb{T}^m)$ for different relations between the parameters $q$, $\tau_1$, $p$, $\tau_2$, $a$, $\theta$. The proofs of the theorems are based on the constructive method developed by V.N. Temlyakov.