Geodesic
Optimal Cubature Formulas on Classes of Periodic Functions in Several Variables
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function Spaces, Approximation Theory, and Related Problems of Analysis, Tome 312 (2021), pp. 22-42.

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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. 
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D. B. Bazarkhanov. Optimal Cubature Formulas on Classes of Periodic Functions in Several Variables. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function Spaces, Approximation Theory, and Related Problems of Analysis, Tome 312 (2021), pp. 22-42. http://geodesic.mathdoc.fr/item/TM_2021_312_a1/

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