Nonlinear approximations of classes of periodic functions of many variables

Order-sharp estimates are established for the best $N$-term approximations of functions in the classes $\mathrm B^{sm}_{pq}(\mathbb T^k)$ and $\mathrm L^{sm}_{pq}(\mathbb T^k)$ of Nikol'skii–Besov and Lizorkin–Triebel types with respect to the multiple system $\widetilde {\mathcal W}^m$ of Meyer wavelets in the metric of $L_r(\mathbb T^k)$ for various relations between the parameters $s,p,q,r$, and $m$ ($s=(s_1,\dots,s_n)\in\mathbb R^n_+$, $1\leq p,q,r\leq\infty$, $m=(m_1,\dots,m_n)\in\mathbb N^n$, and $k=m_1+\dots+m_n$). The proof of upper estimates is based on variants of the so-called greedy algorithms.