Аннотация
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.