Application of the divisors theory to numerical integration of periodic functions in several variables

N. Temirgaliev

Sbornik. Mathematics, Tome 69 (1991) no. 2, pp. 527-542

Résumé

The author exhibits an application of ideal theory to the exact integration of trigonometric polynomials with arbitrarily prescribed coefficient spectrum and to the approximate integration of functions of certain multidimensional classes $E$, $W$, and $H$.

Bibliographie

[1] Korobov N. M., “Priblizhennoe vychislenie kratnykh integralov s pomoschyu metodov teorii chisel”, Dokl. AN SSSR, 115:6 (1957), 1062–1065 | MR | Zbl

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

[3] Korobov N. M., “O vychislenii optimalnykh koeffitsientov”, Dokl. AN SSSR, 267:2 (1982), 289–292 | MR | Zbl

[4] Bakhvalov N. S., “O priblizhennom vychislenii kratnykh integralov”, Vestn. MGU. Ser. 1, Matem., mekh., 1959, no. 4, 3–18 | MR

[5] Bakhvalov N. S., “Ob optimalnykh otsenkakh skhodimosti g kvadraturnykh protsessov i metodov integrirovaniya tipa Monte-Karlo na klassakh funktsii”, Zhurnal vych. matem. i matem. fiziki, 4:4 (1964), 5–63, Dopolnenie. (Chislennye metody $\dots$) | MR | Zbl

[6] Temlyakov V. N., “O vosstanovlenii periodicheskikh funktsii neskolkikh peremennykh po znacheniyam v uzlakh teoretiko-chislovykh setok”, Anal. Math., 12 (1986), 287–305 | DOI | MR | Zbl

[7] Temlyakov V. N., “O priblizhennom vosstanovlenii periodicheskikh funktsii neskolkikh peremennykh”, Dokl. AN SSSR, 280:6 (1985), 1310–1313 | MR | Zbl

[8] Temlyakov V. N., “Kvadraturnye formuly i vosstanovlenie po znacheniyam v uzlakh teoretiko-chislovykh setok dlya klassov funktsii maloi gladkosti”, UMN, 40:4 (1985), 203–204

[9] Temlyakov V. N., “Priblizhennoe vosstanovlenie periodicheskikh funktsii neskolkikh peremennykh”, Matem. sb., 128 (170) (1985), 256–268 | MR | Zbl

[10] Smolyak S. A., “Interpolyatsionnye i kvadraturnye formuly na klassakh $W_s^\alpha$ i $E_s^\alpha$”, Dokl. AN SSSR, 131:5 (1960), 1028–1031

[11] Frolov K. K., “Otsenki sverkhu pogreshnosti kvadraturnykh formul na klassakh funktsii”, Dokl. AN SSSR, 231:4 (1976), 818–821 | MR | Zbl

[12] Bykovskii V. A., O pravilnom poryadke pogreshnosti optimalnykh kvadraturnykh formul v prostranstvakh s dominiruyuschei proizvodnoi i kvadratichnykh otkloneniyakh setok, Preprint VTs, VTs, Vladivostok, 1985 | MR

[13] Hua Loo Keng, Wang Yuan, Applications of Number Theory to Numerical Analysis, Springer Verlag, Berlin, Heidelberg, New York, 1981 | MR

[14] Keipers L., Niderreiter G., Ravnomernoe raspredelenie posledovatelnostei, Nauka, M., 1985 | MR

[15] Hlawka E., Firneis F., Zinterhof P., Zahlentheoretische Methoden in der numerischen Mathematik, Wien, München, Oldenbourg, 1981 | MR

[16] Niederreiter H., “Quasi Monte-Carlo methods and pseudorandoms number”, Amer. Math. Soc., 84:6 (1978), 957–1041 | DOI | MR | Zbl

[17] Voronin S. M., Temirgaliev N., “O kvadraturnykh formulakh, svyazannykh s divizorami polya gaussovykh chisel”, Matem. zametki, 46:2 (1989), 34–41 | MR | Zbl

[18] Borevich Z. I., Shafarevich I. R., Teoriya chisel, Nauka, M., 1985 | MR | Zbl

[19] Veil G., Algebraicheskaya teoriya chisel, IL, M., 1947

[20] Edvards G., Poslednyaya teorema Ferma, Mir, M., 1980 | MR

[21] Karatsuba A. A., Osnovy analiticheskoi teorii chisel, Nauka, M., 1983 | MR

[22] Prakhar K., Raspredelenie prostykh chisel, Mir, M., 1967 | MR

[23] Temlyakov V. N., Priblizhenie funktsii s ogranichennoi smeshannoi proizvodnoi, Tr. MIAN SSSR, 178, 1986 | MR | Zbl