Computational (Numerical) diameter in a context of general theory of a recovery

N. Temirgaliev ; A. Zh. Zhubanysheva

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2019), pp. 89-97.

Résumé

We discuss a C(N)D-statement consisting of the known and elaborating in decades C(N)D-1 statement which can be and should be interpreted as quantitative statement of approximation theory and calculus mathematics, which together with new prolongations of C(N)D-2 and -3 in aggregate is suggested as natural theoretical and computational scheme of further developments of numerical analysis.

Détail

Keywords: computational (Numerical) Diameter (C(N)D), approximation theory in quantitative statement, calculus mathematics, recovery by exact and inexact information, limiting error, new scheme of numerical analysis.

Bibliographie

[1] Temirgaliev N., “Teoretiko-chislovye metody i teoretiko-veroyatnostnyi podkhod k zadacham analiza. Teoriya vlozhenii i priblizhenii, absolyutnaya skhodimost i preobrazovaniya ryadov Fure”, Vestn. EAU, 3 (1997), 90–144 | Zbl

[2] Kolmogorov A. N., “Über die beste Annäherung von Funktionen einer gegebenen Funktionen klasse”, Ann. Math., 37:1 (1936), 107–110 | DOI | MR | Zbl

[3] Stechkin S. B., “O nailuchshem priblizhenii zadannykh klassov funktsii lyubymi polinomami”, UMN, 9:1 (1954), 133–134 | Zbl

[4] Osipenko K. Yu., “Nailuchshee priblizhenie analiticheskikh funktsii po informatsii ob ikh znacheniyakh v konechnom chisle tochek”, Matem. zametki, 19:1 (1976), 29–40 | Zbl

[5] Heinrich S., “Random approximation in numerical analysis”, Func. Anal., eds. K.D. Bierstedt, A. Pietsch, W.M. Ruess, D. Vogt, Marcel Dekker, New York, 1993, 123–171 | MR

[6] Khardi G., Raskhodyaschiesya ryady, In. lit., M., 1951

[7] Volkov I. I., Ulyanov P. L., “O nekotorykh novykh rezultatakh ob obschei teorii summirovaniya ryadov i posledovatelnostei”: R. Kuk, Beskonechnye matritsy i prostranstva posledovatelnostei, GIFML, M., 1960, 363–467

[8] Kashin B. S., Saakyan A. A., Ortogonalnye ryady, AFTs, M., 1999

[9] Korneichuk N. P., Tochnye konstanty v teorii priblizheniya, Nauka, M., 1987

[10] Kashin B. S., Kulikova T. Yu., “O spravedlivosti dlya freimov odnogo rezultata ob ortogonalnykh sistemakh”, Matem. zametki, 77:2 (2005), 311–312 | DOI | MR | Zbl

[11] Blatter K., Veivlet-analiz. Osnovy teorii, Tekhnosfera, M., 2004

[12] Novikov I. Ya., Protasov V. Yu., Skopina M. A., Teoriya vspleskov, Fizmatlit, M., 2005

[13] Temlyakov V. N., Approximation of periodic functions, Comput. math. and anal. ser., Nova Sci. Publ., Commack, NY, 1993 | MR | Zbl

[14] Tikhomirov V. M., Nekotorye voprosy teorii priblizhenii, Izd-vo MGU, M., 1976

[15] Zigmund A., Trigonometricheskie ryady, Mir, M., 1965 | MR

[16] Ryabenkii V. S., Vvedenie v vychislitelnuyu matematiku, Nauka, M., 1994

[17] Temlyakov V. N., Greedy approximation, Cambridge University Press, Cambridge, 2011 | MR | Zbl

[18] Sard A., “Best approximate integration formulas; best approximation formulas”, Amer. J. Math., 71 (1949), 80–91 | DOI | MR | Zbl

[19] Nikolskii S. M., “K voprosu ob otsenkakh priblizhenii kvadraturnymi formulami”, UMN, 5:2(36) (1950), 165–177 | Zbl

[20] Bakhvalov N. S., “O priblizhennom vychislenii kratnykh integralov”, Vestn. MGU. Ser. matem. mekhan., 4 (1959), 3–18

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

[22] Sharygin I. F., “Otsenki snizu pogreshnosti kvadraturnykh formul na klassakh funktsii”, Zhurn. vychisl. matem. i matem. fiz., 3:3 (1963), 370–376 | Zbl

[23] Babushka I., Sobolev S., “Optimizatsiya chislennykh metodov”, Apl. Matem., 10 (1965), 96–129 | MR | Zbl

[24] Ioffe A. D., Tikhomirov V. M., “Dvoistvennost vypuklykh funktsii i ekstremalnye zadachi”, UMN, 23:6(144) (1968), 51–116 | Zbl

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

[26] Micchelli C. A. and Rivlin T. J., “A survey of optimal recovery”, Optimal estimation in approximation theory, eds. C.A. Micchelli, T. J. Rivlin, Plenum Press, New York, 1977, 1–54 | MR

[27] Fisher S. D., Micchelli Ch. A., “Optimal sampling of holomorphic functions”, Amer. J. Math., 106:3 (1984), 593–609 | DOI | MR | Zbl

[28] Pietsch A., Eigenvalues and $s$-numbers, Geest and Portig, Leipzig; Cambridge Univ. Press, 1987 | MR | Zbl

[29] Traub J. F., Wasilkowski G. W. and Wozniakowski H., Information-based complexity, Academic Press, New York, 1988 | MR | Zbl

[30] Novak E., Wozniakowski H., Tractability of multivariate problems, v. 1, EMS Tracts Math., 6, Linear information, European Math. Soc. Publishing House, Zurich, 2008 | MR | Zbl

[31] Lokutsievskii O. V., Gavrikov M. B., Nachala chislennogo analiza, TOO «Yanus», M., 1995

[32] Plaskota L., Noisy information and computational complexity, Cambridge University Press, 1996 | MR | Zbl

[33] Magaril-Ilyaev G. G., Osipenko K. Yu., “Optimalnoe vosstanovlenie znachenii funktsii i ikh proizvodnykh po netochno zadannomu preobrazovaniyu Fure”, Matem. sb., 195:10 (2004), 67–82 | DOI | Zbl

[34] Marchuk A. G., Osipenko K. Yu., “Nailuchshee priblizhenie funktsii, zadannykh s pogreshnostyu v konechnom chisle tochek”, Matem. zametki, 17:3 (1975), 359–368 | Zbl

[35] Azhgaliev Sh., Temirgaliev N., “Ob informativnoi moschnosti lineinykh funktsionalov”, Matem. zametki, 3:6 (2003), 803–812 | DOI

[36] Temirgaliev N., Zhubanysheva A. Zh., “Poryadkovye otsenki norm proizvodnykh funktsii s nulevymi znacheniyami na lineinykh funktsionalakh i ikh primeneniya”, Izv. vuzov. Matem., 2017, no. 3, 89–95

[37] Zhubanysheva A. Zh., Temirgaliev N., “Informativnaya moschnost trigonometricheskikh koeffitsientov Fure i ikh predelnaya pogreshnost pri diskretizatsii operatora differentsirovaniya na mnogomernykh klassakh Soboleva”, Zhurn. vychisl. matem. i matem. fiz., 55:9 (2015), 1474–1485 | DOI | MR | Zbl

[38] Temirgaliev N., Sherniyazov K. E., Berikkhanova M. E., “Tochnye poryadki kompyuternykh (vychislitelnykh) poperechnikov v zadachakh vosstanovleniya funktsii i diskretizatsii reshenii uravneniya Kleina–Gordona po koeffitsientam Fure”, Sovremen. probl. matem., 17 (2013), 179–207 | DOI

[39] Temirgaliev N., Abikenova Sh. K., Zhubanysheva A. Zh., Taugynbaeva G. E., “Zadachi diskretizatsii reshenii volnovogo uravneniya, chislennogo differentsirovaniya i vosstanovleniya funktsii v kontekste kompyuternogo (vychislitelnogo) poperechnika”, Izv. vuzov. Matem., 2013, no. 8, 86–93 | Zbl

[40] Abikenova Sh., Utesov A., Temirgaliev N., “O diskretizatsii reshenii volnovogo uravneniya s nachalnymi usloviyami iz obobschennykh klassov Soboleva”, Matem. zametki, 91:3 (2012), 459–463 | DOI | Zbl

[41] Temirgaliev N., Kudaibergenov S. S., Shomanova A. A., “Primeneniya kvadraturnykh formul Smolyaka k chislennomu integrirovaniyu koeffitsientov Fure ivz adachakh vosstanovleniya”, Izv. vuzov. Matem., 2010, no. 3, 52–71 | Zbl

[42] Ibatulin I., Temirgaliev N., “Ob informativnoi moschnosti vsekh vozmozhnykh lineinykh funktsionalov pri diskretizatsii reshenii uravneniya Kleina–Gordona v metrike $L^{2,\infty}$”, Differents. uravneniya, 44:4 (2008), 491–506 | Zbl

[43] Temirgaliev N., “Tenzornye proizvedeniya funktsionalov i ikh primeneniya”, Dokl. RAN, 430:4 (2010), 460–465 | Zbl

[44] Nauryzbaev N. Zh., Shomanova A. A., Temirgaliev N., “O nekotorykh osobykh effektakh v teorii chislennogo integrirovaniya i vosstanovleniya funktsii”, Izv. vuzov. Matem., 2018, no. 3, 96–102