On Bykovsky estimates for deviations of generalized parallelepipedal grids
https://doi.org/10.22405/2226-8383-2023-24-2-214-227
Abstract
This paper is devoted to obtaining estimates of the type of Bykovsky estimates for the deviation of a generalized parallelepipedal grid. It continues the studies similar to those that we previously performed to assess the quality measure and the quantitative measure of the
parallelepipedal grid.
The main idea used in this paper goes back to the work of V. A. Bykovsky (2002) on estimating the error of approximate integration over parallelepipedal grids and its generalization in the work of O. A. Gorkusha and N. M. Dobrovolsky (2005) for the case of a hyperbolic zeta function of an arbitrary lattice. The central place in these works is played by the Bykovsky set, consisting of local minima of the second kind, and sums over these sets.
As in the work "On Bykovsky estimates for a measure of the quality of optimal coefficients the effect was found that a multiplier with a logarithmic order of growth appears in the deviation estimates, which began to include the definition of the modified Bykovsky sum.
The method of work is to combine the approaches from the work "Estimates of deviations of generalized parallelepipedal grids"(1984) with the approaches of 2005.
Further ways to obtain clarification of the received estimates are outlined.
Keywords
quality function,
generalized parallelepipedal grid,
Bykovsky set,
Bykovsky sum,
local lattice minima,
minimal comparison solutions.
Supporting agencies: Acknowledgments: The reported study was funded by the RSF No. 23-21-00317 on the topic “Number geometry and Diophantine approximations in the number-theoretic method in approximate analysis”.
About the Authors
Antonina Nikolaevna KormachevaSwitzerland
Nikolai Nikolaevich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Irina Yuryevna Rebrova
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Nikolai Mihailovich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
References
1. Bakhvalov, N.S. 1959, “On approximate computation of multiple integrals”, Vestnik Moskovskogo universiteta, no. 4, pp. 3–18.
2. Bykovskij, V.А 2002, “On the error of number-theoretic quadrature formulas”, Chebyshevskij sbornik, vol. 3, no. 2(4), pp. 27–33.
3. O. A. Gorkusha, N. M. Dobrovolsky, 2005, "On estimates of hyperbolic zeta function of lattices" // Chebyshevsky Collection, vol. 6, issue 2(14), pp. 130-138.
4. Dobrovol’skii, N. M. 1984, “The hyperbolic Zeta function of lattices”, Dep. v VINITI, no. 6090–84.
5. Dobrovol’skii, N. M. 1984, “Evaluation of generalized variance parallelepipedal grids”, Dep. v VINITI, no. 6089–84.
6. N. N. Dobrovol’skii, M. N. Dobrovol’skii, I. Yu. Rebrova, N. M. Dobrovol’skii, 2022, "The final deviation and the main quality measure for Korob ov grids Chebyshevskii sbornik, vol. 23, no. 2, pp. 56–73.
7. Dobrovol’skii, N. M., Esayan, А.R., Pikhtil’kov, S.А., Rodionova, O.V. & Ustyan, А.E. 1999, “On a single algorithm for finding optimal coefficients”, Izvestiya TulGU. Seriya Matematika. Mekhanika. Informatika, vol. 5, no. 1, pp. 51–71.
8. Dobrovol’skii, N. M., Esayan, А.R. & Rebrova, I. YU. 1998, “On a recursive algorithm for lattices”, Teoriya priblizhenij i garmonicheskij analiz: Tezisy doklada Mezhdunarodnoj konferentsii (Approximation theory and harmonic analysis: proceedings of the International conference), Tula, Russia.
9. Dobrovol’skii, N. M., Esayan, А.R. & Rebrova, I. YU. 1998, “On a recursive algorithm for lattices”, Izvestiya TulGU. Seriya Matematika. Mekhanika. Informatika, vol. 5, no. 3, pp. 38–51.
10. Dobrovol’skii, N. M. & Korobov, N. M. 2002, “On the error estimation of quadrature formulas with optimal parallelepipedal grids”, Chebyshevskij sbornik, vol. 3, no. 1(3), pp. 41–48.
11. A. N. Kormacheva, N. N. Dobrovol’skii, N. M. Dobrovol’skii, 2021, “On the hyp erb olic parameter of a two-dimensional lattice of comparisons”, Chebyshevskii sbornik, vol. 22, no. 4, pp. 168–182.
12. Korobov, N.M. 1959, “The evaluation of multiple integrals by method of optimal coefficients”, Vestnik Moskovskogo universiteta, no. 4, pp. 19–25.
13. Korobov, N.M. 1960, “Properties and calculation of optimal coefficients”, Doklady Аkademii nauk SSSR, vol. 132, no. 5, pp. 1009–1012.
14. Korobov, N.M. 2004, Teoretiko-chislovye metody v priblizhennom analize [Number-theoretic methods in approximate analysis], 2nd ed, MTSNMO, Moscow, Russia.
15. Mikhlyaeva, A. V., 2018, "Approximation of quadratic algebraic lattices and nets by integer lattices and rational nets" , Chebyshevskii sbornik, vol. 19, no. 3, pp. 241–256.
16. Mikhlyaeva, A. V., 2019, "Quality function for the approximation of quadratic algebraic nets", Chebyshevskii sbornik, vol. 20, no. 1, pp. 307–312.
17. Seregina N. K., 2013, "Algorithms of numerical integration with the stopping rule" , TulSU extraction. Natural sciences. Issue 3. pp. 193 — 201.
18. Seregina N. K., 2015, "On the quantitative measure of the quality of optimal coefficients", Izvestiya TulSU. Natural sciences. Issue 1, pp. 22–29.
Review
For citations:
Kormacheva A.N., Dobrovol’skii N.N., Rebrova I.Yu., Dobrovol’skii N.M. On Bykovsky estimates for deviations of generalized parallelepipedal grids. Chebyshevskii Sbornik. 2023;24(2):214-227. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-2-214-227
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