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Optimal Cubature Formulas on Classes of Periodic Functions in Several Variables

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Abstract

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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Notes

  1. Here and below \(\sum\equiv\sum_{\xi\in{\mathbb Z}^m}\), \(\sum^\prime\equiv\sum_{\xi\in{\mathbb Z}^m\setminus\{ {\mathbf 0} \}}\), and \(\sum_{ {\mathtt{k}} }\equiv\sum_{ {\mathtt{k}} \in{\mathbb N}_0^n}\).

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Funding

The work was supported by the Ministry of Education and Science, Republic of Kazakhstan, under grant AP05133257.

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Authors and Affiliations

  1. Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science of the Republic of Kazakhstan, Pushkina Str. 125, Almaty, 050010, Kazakhstan

    D. B. Bazarkhanov

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Correspondence to D. B. Bazarkhanov.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 312, pp. 22–42 https://doi.org/10.4213/tm4153.

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Bazarkhanov, D.B. Optimal Cubature Formulas on Classes of Periodic Functions in Several Variables. Proc. Steklov Inst. Math. 312, 16–36 (2021). https://doi.org/10.1134/S0081543821010028

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  • DOI: https://doi.org/10.1134/S0081543821010028

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