Optimal Cubature Formulas on Classes of Periodic Functions in Several Variables

We establish sharp order estimates for the error of optimal cubature formulas on the Nikol'skii–Besov and Lizorkin–Triebel type spaces, $B^{s\,\mathtt {m}}_{p\,q}(\mathbb T^m)$ and $L^{s\,\mathtt {m}}_{p\,q}(\mathbb T^m)$, respectively, for a number of relations between the parameters $s$, $p$, $q$, and $\mathtt {m}$ ($s=(s_1,\dots ,s_n)\in \mathbb R^n_+$, $1\leq p,q\leq \infty $, $\mathtt {m}=(m_1,\dots ,m_n)\in \mathbb N ^n$, $m=m_1+\dots +m_n$). Lower estimates are proved via Bakhvalov's method. Upper estimates are based on Frolov's cubature formulas.