Nonlinear trigonometric approximations of multivariate function classes

Order-sharp estimates are established for the best $N$-term approximations of functions from Nikol'skii–Besov type classes $\mathrm B^{sm}_{pq}(\mathbb T^k)$ with respect to the multiple trigonometric system $\mathfrak T^{(k)}$ in the metric of $L_r(\mathbb T^k)$ for a number of relations between the parameters $s,p,q,r$, and $m$ ($s=(s_1,\dots,s_n)\in\mathbb R^n_+$, $1\leq p,q,r\leq\infty$, $m=(m_1,\dots,m_n)\in\mathbb N^n$, $k=m_1+\dots+m_n$). Constructive methods of nonlinear trigonometric approximation –variants of the so-called greedy algorithms – are used in the proofs of upper estimates.