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 Chebyshevskii Sb., 2019, Volume 20, Issue 3, Pages 193–219 (Mi cheb807)

About three-dimensional nets of Smolyak I

N. N. Dobrovol'skiiab, D. V. Gorbachevb, V. I. Ivanovb

a Tula State L. N. Tolstoy Pedagogical University (Tula)
b Tula State University (Tula)

Abstract: The work refers to the analytical theory of numbers and it deals with the application of number theory to problems of approximate analysis.
The concept of the hyperbolic parameter of grids with weights and the analogue of Bakhvalov's theorem for the hyperbolic parameter of grids with weights and the hyperbolic Zeta function of grids are considered.
In this paper the following results are obtained:
• a strengthened generalized Bakhvalov–Korobov theorem for the hyperbolic Zeta function of three-dimensional grids is proved;
• the number of nodes of the resin grid is calculated taking into account their multiplicity; the number of nodes taking into account their weights.
• the number of nodes of the resin grid is calculated without taking into account their multiplicity;
• the number of nodes of the resin grid is calculated taking into account their weights;
• the form of a quadrature formula with a resin grid without multiple nodes is found and explicit formulas for the weights of this quadrature formula are found. It is shown that the number of nodes of such a quadrature formula is 7 times less than in the case of a formula with multiple nodes.

Keywords: grid Smolyak, quadrature formulas with grids of Smolyak, interpolation formula with grids of Smolyak.

 Funding Agency Grant Number Russian Foundation for Basic Research 19-41-710005_ð_à Acknowledgments: The reported study was funded by RFBR, project number 19-41-710005_r_a.

DOI: https://doi.org/10.22405/2226-8383-2018-20-3-193-219

Full text: PDF file (789 kB)
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UDC: 511.3
Accepted:12.11.2019

Citation: N. N. Dobrovol'skii, D. V. Gorbachev, V. I. Ivanov, “About three-dimensional nets of Smolyak I”, Chebyshevskii Sb., 20:3 (2019), 193–219

Citation in format AMSBIB
\Bibitem{DobGorIva19} \by N.~N.~Dobrovol'skii, D.~V.~Gorbachev, V.~I.~Ivanov \paper About three-dimensional nets of Smolyak I \jour Chebyshevskii Sb. \yr 2019 \vol 20 \issue 3 \pages 193--219 \mathnet{http://mi.mathnet.ru/cheb807} \crossref{https://doi.org/10.22405/2226-8383-2018-20-3-193-219}