The final deviation and the main quality measure for Korobov grids

N. N. Dobrovol'skii ; M. N. Dobrovol'skii ; I. Yu. Rebrova ; N. M. Dobrovol'skii

Čebyševskij sbornik, Tome 23 (2022) no. 2, pp. 56-73.

Résumé

The paper considers four new concepts: a modified basic measure of the quality of a set of coefficients, absolutely optimal coefficients of the index $s$, the mathematical expectation of the local deviation of the parallelepipedal grid and the variance of the local deviation of the parallelepipedal grid. It is shown that at least $\frac{(p-1)^s}{2}$ of different sets $(a_1,\ldots,a_s)$ integers mutually prime with the module $p$ will be absolutely optimal sets of the index $s$ with the constant $B=2s$. It is established that any absolutely optimal set of optimal coefficients of the $s$ index is an optimal set of optimal coefficients of the $s$ index, while any subset of its $s_1$ coefficients is an optimal set of optimal coefficients of the $s_1$ index. For the finite deviation introduced by N. M. Korobov in 1967, new formulas and estimates are obtained for parallelepipedal grids. In this paper, for the first time, the concept of the mathematical expectation of a local deviation is considered and a convenient formula for its calculation is found. The concept of local deviation variance is also considered for the first time. The paper outlines the directions of further research on this topic.

Détail

Keywords: finite deviation, the main measure of quality, Korobov grids, finite Fourier series.

Bibliographie

[1] Avdeeva M. O., “Otsenka kolichestva lokalnykh minimumov tselochislennykh reshetok”, Chebyshevskii sbornik, 5:4(12) (2004), 35–38 | MR | Zbl

[2] Babenko K. I., Osnovy chislennogo analiza, Nauka, M., 1986

[3] Bykovskii V. A., “O pogreshnosti teoretiko-chislovykh kvadraturnykh formul”, Chebyshevskii sbornik, 3:2(4) (2002), 27–33 | MR | Zbl

[4] Bykovskii V. A., “O pogreshnosti teoretiko-chislovykh kvadraturnykh formul”, Dokl. RAN, 389:2 (2003), 154–155 | MR | Zbl

[5] Gorkusha O. A., “Kriterii konechnosti mnozhestva lokalnykh minimumov reshetki”, Algebra i teoriya chisel: sovremennye problemy i prilozheniya, Tez. dokl. VI Mezhdunar. konf., posvyaschennoi 100-letiyu N. G. Chudakova (Saratov, 13 — 17 sentyabrya 2004 g.), Izd-vo Sarat. un-ta, Saratov, 2004, 47

[6] Gorkusha O. A., “Kriterii konechnosti mnozhestva lokalnykh minimumov reshetki”, Chebyshevskii sbornik, 5:3(11) (2002), 15–17 | MR

[7] Demidov S. S., Morozova E. A., Chubarikov V. N., Rebrova I. Yu., Balaba I. N., Dobrovolskii N. N., Dobrovolskii N. M., Dobrovolskaya L. P., Rodionov A. V., Pikhtilkova O. A., “Teoretiko-chislovoi metod v priblizhennom analize”, Chebyshevskii sb., 18:4 (2017), 6–85 | DOI | MR | Zbl

[8] Dobrovolskii N. M., Giperbolicheskaya dzeta funktsiya reshetok, Dep. v VINITI 24.08.84, No 6090–84

[9] A. N. Kormacheva, N. N. Dobrovolskii, I. Yu. Rebrova, N. M. Dobrovolskii, “O giperbolicheskom parametre dvumernoi reshetki sravnenii”, Chebyshevckii sbornik, 22:4 (2021), 168–182 | MR

[10] Korobov N. M., “O priblizhennom vychislenii kratnykh integralov”, DAN SSSR, 124:6 (1959), 1207–1210 | Zbl

[11] Korobov N. M., “Vychislenie kratnykh integralov metodom optimalnykh koeffitsientov”, Vestn. Mosk. un-ta, 1959, no. 4, 19–25

[12] Korobov N. M., “Svoistva i vychislenie optimalnykh koeffitsientov”, DAN SSSR, 132:5 (1960), 1009–1012 | Zbl

[13] Korobov N. M., Teoretiko-chislovye metody v priblizhennom analize, Fizmatgiz, M., 1963

[14] Korobov N. M., “O nekotorykh voprosakh teorii diofantovykh priblizhenii”, UMN, 22:3 (135) (1967), 83–118 | MR | Zbl

[15] Korobov N. M., Teoretiko-chislovye metody v priblizhennom analize, vtoroe izdanie, MTsNMO, M., 2004

[16] O. V. Lokutsievskii, M. B. Gavrikov, Nachala chislennogo analiza, TOO “Yanus”, M., 1995 | MR

[17] Roth K. F., “On irregularities of distribution”, Mathematika, 1954, no. 1, 73–79 | DOI | MR | Zbl