Geodesic
About three-dimensional nets of Smolyak I
Čebyševskij sbornik, Tome 20 (2019) no. 3, pp. 193-219.

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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. 
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N. N. Dobrovol'skii; D. V. Gorbachev; V. I. Ivanov. About three-dimensional nets of Smolyak I. Čebyševskij sbornik, Tome 20 (2019) no. 3, pp. 193-219. http://geodesic.mathdoc.fr/item/CHEB_2019_20_3_a13/

[1] Bakhvalov N. S., “On approximate computation of multiple integrals”, Vestnik Moskovskogo universiteta, 1959, no. 4, 3–18

[2] Vronskaya G. T., Dobrovol'skii N. N., Standard deviation of a flat mesh, Izdatel'stvo TGPU im L.N. Tolstogo, Tula, Russia, 2012

[3] Dobrovol'skaya L. P., Dobrovol'skii M. N., Dobrovol'skii N. M., Dobrovol'skii N. N., Multidimensional number-theoretic grids and lattices and algorithms for finding optimal coefficients, Izdatel'stvo Tul'skogo gosudarstvennogo pedagogicheskogo universiteta im. L.N. Tolstogo, Tula, Russia, 2012, 284 pp.

[4] Dobrovol'skaya L. P., Dobrovol'skii M. N., Dobrovol'skii N. M., Dobrovol'skii N. N., “The hyperbolic Zeta function of grids and lattices, and calculation of optimal coefficients”, Chebyshevskij sbornik, 13:4 (44) (2012), 4–107 | Zbl

[5] Dobrovol'skaya L. P., Dobrovol'skii N. M., Dobrovol'skii N. N., Ogorodnichuk N. K., Rebrov E. D., Rebrova I. YU., “Some questions of the number-theoretic method in the approximate analysis”, Scientific notes of Orel state University, 2012, no. 6-2, Proceedings of the X international conference “Algebra and number theory: modern problems and applications”, 90–98

[6] Dobrovol'skaya L. P., Dobrovol'skii M. N., Dobrovol'skii N. M., Dobrovol'skii N. N., Rebrova I. YU., “Some questions of the number-theoretic method in the approximate analysis”, Izvestie Saratovskogo universiteta. Novaya seriya. Seriya: Matematika. Mekhanika. Informatika, 2013, no. 4 (2), 47–52 | Zbl

[7] Dobrovol'skaya L. P., Dobrovol'skii N. M., Simonov A. S., “On the error of approximate integration over modified grids”, Chebyshevskij sbornik, 9:1 (25) (2008), 185–223 | MR | Zbl

[8] Dobrovol'skii M. N., “Estimates of sums over a hyperbolic cross”, Izvestiya Tul'skogo gosudarstvennogo universiteta. Seriya: Matematika. Mekhanika. Informatika, 9:1 (2003), 82–90 | MR

[9] Dobrovol'skii N. M., The hyperbolic Zeta function of lattices, Dep. v VINITI, No 6090-84, 1984

[10] Dobrovol'skii N. M., Evaluation of generalized variance parallelepipedal grids, Dep. v VINITI, No 6089-84, 1984

[11] Dobrovol'skii N. M., On quadrature formulas in classes $E_s^\alpha(c)$ and $H_s^\alpha(c)$, Dep. v VINITI, No 6091-84, 1984

[12] Dobrovol'skii N. M., Esayan A. R., Yafaeva R. R., “On grids of Smolyak S A.”, Sovremennye problemy matematiki, mekhaniki, informatiki, Tezisy dokladov Vserossijskoj nauchnoj konferentsii (Tula, Russia, 2002), 18–20

[13] Dobrovol'skii N. M., Manokhin E. V., “Banach spaces of periodic functions”, Izvestiya TulGU. Seriya Matematika. Mekhanika. Informatika, 4:3 (1998), 56–67 | MR

[14] Dobrovol'skii N. M., Manokhin E. V., Rebrova I. YU., Roshhenya A. L., “On the continuity of the Zeta function of a grid with weights”, Izvestiya TulGU. Seriya Matematika. Mekhanika. Informatika, 7:1 (2001), 82–86 | MR

[15] Dobrovol'skii N. N., “On the number of integer points in a hyperbolic cross at the values of $1\leqslant t21$”, Izvestiya TulGU. Seriya Matematika. Mekhanika. Informatika, 9:1 (2003), 91–95 | MR

[16] Dobrovol'skii N. N., “Deviation of two-dimensional Smolyak grids”, Chebyshevskij sbornik, 8:1 (21) (2007), 110–152 | MR | Zbl

[17] Dobrovol'skii N. N., “A trigonometric polynomial on a grid of Smolyak”, Modern problems of mathematics, mechanics, computer science, Proceedings of the international scientific conference (Tula, Russia, 2007), 34–36

[18] Dobrovol'skii N. N., Izvestiya TulGU. Seriya Matematika. Mekhanika. Informatika, 2013, no. 2-1, 6–18

[19] Dobrovol'skii N. N., Hyperbolic parameter of meshes with weights and its application, Ph.D. Thesis, Moscow State University, Moscow, Russia, 2014

[20] O. V. Kiseleva, “On the Korobov problem for modified resin grids”, Chebyshevskij sbornik, 8:4 (24) (2007), 50–104 | Zbl

[21] Korobov N. M., “Approximate evaluation of multiple integrals by using methods of the theory of numbers”, Doklady Akademii nauk SSSR, 1957, no. 6, 1062–1065 | Zbl

[22] Korobov N. M., “The evaluation of multiple integrals by method of optimal coefficients”, Vestnik Moskovskogo universiteta, 1959, no. 4, 19–25

[23] Korobov N. M., “On approximate computation of multiple integrals”, Doklady Akademii nauk SSSR, 124:6 (1959), 1207–1210 | Zbl

[24] Korobov N. M., “Properties and calculation of optimal coefficients”, Doklady Akademii nauk SSSR, 132:5 (1960), 1009–1012 | Zbl

[25] Korobov N. M., Number-theoretic methods in approximate analysis, Fizmatgiz, Russia, 1963

[26] Korobov N. M., “Quadrature formulas with combined grids”, Matematicheskie zametki, 55:2 (1994), 83–90 | MR | Zbl

[27] Korobov N. M., Number-theoretic methods in approximate analysis, 2nd ed, MTSNMO, Russia, 2004

[28] I. Yu. Rebrova, V. N. Chubarikov, N. N. Dobrovol'skii, M. N. Dobrovol'skii, N. M. Dobrovol'skii, “On classical number-theoretic nets”, Chebyshevskii sbornik, 20:3 (2018), 118–176 | MR | Zbl

[29] Smolyak S. A., “Quadrature and interpolation formulas on tensor products of some classes of functions”, Doklady Akademii nauk SSSR, 148:5 (1963), 1042–1045 | Zbl

[30] Sobol' I. M., Multidimensional quadrature formulas and Haar functions, Nauka, USSR, 1969

[31] Frolov K. K., “Upper bounds on the error of quadrature formulas on classes of functions”, Doklady Akademii nauk SSSR, 231:4 (1976), 818–821 | MR | Zbl

[32] Frolov K. K., Quadrature formulas on classes of functions, Ph. D. Thesis, Vychislitel'nyj tsentr Akademii Nauk SSSR, Moscow, USSR, 1979

[33] Chandrasekharan K., Vvedenie v analiticheskuju teoriju chisel, Izd-vo Mir, M., 1974, 188 pp.

[34] Dobrovolskaya L. P., Dobrovolsky M. N., Dobrovol'skii N. M., Dobrovolsky N. N., “On Hyperbolic Zeta Function of Lattices”, Continuous and Distributed Systems, Solid Mechanics and Its Applications, 211, 2014, 23–62 | DOI | MR | Zbl

[35] H. Faure, “Discrepance de suites associees a un systeme denumeration (en dimention s)”, Acta Arith., 41 (1982), 337–351 | MR | Zbl

[36] J. H. Halton, “On the efficiency of certain quasirandom sequences of points in evaluating multidimensional integrals”, Numerische Math., 27:2 (1960), 84–90 | MR

[37] J. M. Hammersley, “Monte-Carlo methods for sobving multivariable problems”, Proc. Acad. Sci., 86:4 (1960), 844–874 | MR | Zbl

[38] H. Weyl, “Uber die Gleichverteilung von Zahlen mod. Eins”, Math. Ann., 77 (1916), 313–352 ; Veil G., Matematika. Teoreticheskaya fizika, Nauka, M., 1984 | MR | Zbl