Geodesic
Best Polynomial Approximations in~$L_2$ of Classes of $2\pi$-Periodic Functions and Exact Values of Their Widths
Matematičeskie zametki, Tome 90 (2011) no. 5, pp. 764-775.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the problem of determining sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity of $m$th order in the space $L_2$ as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of $n$-widths in $L_2$.
Keywords: best polynomial approximation, periodic differentiable function, trigonometric polynomial, modulus of continuity, the space $L_2$, $n$-width, Fourier series. 
@article{MZM_2011_90_5_a10,
     author = {M. Sh. Shabozov and G. A. Yusupov},
     title = {Best {Polynomial} {Approximations} in~$L_2$ of {Classes} of $2\pi${-Periodic} {Functions} and {Exact} {Values} of {Their} {Widths}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {764--775},
     publisher = {mathdoc},
     volume = {90},
     number = {5},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_90_5_a10/}
}
TY  - JOUR
AU  - M. Sh. Shabozov
AU  - G. A. Yusupov
TI  - Best Polynomial Approximations in~$L_2$ of Classes of $2\pi$-Periodic Functions and Exact Values of Their Widths
JO  - Matematičeskie zametki
PY  - 2011
SP  - 764
EP  - 775
VL  - 90
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2011_90_5_a10/
LA  - ru
ID  - MZM_2011_90_5_a10
ER  - 
%0 Journal Article
%A M. Sh. Shabozov
%A G. A. Yusupov
%T Best Polynomial Approximations in~$L_2$ of Classes of $2\pi$-Periodic Functions and Exact Values of Their Widths
%J Matematičeskie zametki
%D 2011
%P 764-775
%V 90
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2011_90_5_a10/
%G ru
%F MZM_2011_90_5_a10
M. Sh. Shabozov; G. A. Yusupov. Best Polynomial Approximations in~$L_2$ of Classes of $2\pi$-Periodic Functions and Exact Values of Their Widths. Matematičeskie zametki, Tome 90 (2011) no. 5, pp. 764-775. http://geodesic.mathdoc.fr/item/MZM_2011_90_5_a10/

[1] N. I. Chernykh, “O nailuchshem priblizhenii periodicheskikh funktsii trigonometricheskimi polinomami v $L_2$”, Matem. zametki, 2:5 (1967), 513–522 | MR | Zbl

[2] N. I. Chernykh, “O neravenstve Dzheksona v $L_2$”, Priblizhenie funktsii v srednem, Sbornik rabot, Tr. MIAN SSSR, 88, 1967, 71–74 | MR | Zbl

[3] L. V. Taikov, “Neravenstva, soderzhaschie nailuchshie priblizheniya i modul nepreryvnosti funktsii iz $L_2$”, Matem. zametki, 20:3 (1976), 433–438 | MR | Zbl

[4] L. V. Taikov, “Nailuchshie priblizheniya differentsiruemykh funktsii v metrike prostranstva $L_2$”, Matem. zametki, 22:4 (1977), 535–542 | MR | Zbl

[5] L. V. Taikov, “Strukturnye i konstruktivnye kharakteristiki funktsii iz $L_2$”, Matem. zametki, 25:2 (1979), 217–223 | MR | Zbl

[6] A. A. Ligun, “Nekotorye neravenstva mezhdu nailuchshimi priblizheniyami i modulyami nepreryvnosti v prostranstve $L_2$”, Matem. zametki, 24:6 (1978), 785–792 | MR | Zbl

[7] V. V. Shalaev, “O poperechnikakh v $L_2$ klassov differentsiruemykh funktsii, opredelyaemykh modulyami nepreryvnosti vysshikh poryadkov”, Ukr. matem. zhurn., 43:1 (1991), 125–129 | MR | Zbl

[8] S. B. Vakarchuk, “O nailuchshikh polinomialnykh priblizheniyakh v $L_2$ nekotorykh klassov $2\pi$-periodicheskikh funktsii i tochnykh znacheniyakh ikh $n$-poperechnikov”, Matem. zametki, 70:3 (2001), 334–345 | MR | Zbl

[9] S. B. Vakarchuk, “Tochnye konstanty v neravenstvakh tipa Dzheksona i tochnye znacheniya poperechnikov funktsionalnykh klassov iz $L_2$”, Matem. zametki, 78:5 (2005), 792–796 | MR | Zbl

[10] S. B. Vakarchuk, “Neravenstva tipa Dzheksona i poperechniki klassov funktsii v $L_2$”, Matem. zametki, 80:1 (2006), 11–19 | MR | Zbl

[11] N. Ainulloev, “O poperechnikakh differentsiruemykh funktsii v $L_2$”, Dokl. AN TadzhSSR, 28:6 (1985), 309–313 | MR | Zbl

[12] M. Sh. Shabozov, O. Sh. Shabozov, “O poperechnikakh klassov periodicheskikh funktsii v prostranstve $L_2[0,2\pi]$”, Dokl. AN RT, 49:2 (2006), 111–115

[13] Kh. Yussef, “O nailuchshikh priblizheniyakh funktsii i znacheniyakh poperechnikov klassov funktsii v $L_2$”, Primenenie funktsionalnogo analiza v teorii priblizhenii, Sb. nauchn. trudov, Kalininskii gos. un-t, Kalinin, 1988, 100–114 | MR

[14] G. A. Yusupov, “O tochnykh znacheniyakh poperechnikov nekotorykh klassov periodicheskikh differentsiruemykh funktsii v prostranstve $L_2[0,2\pi]$”, Dokl. AN RT, 51:12 (2008), 810–817

[15] V. A. Yudin, “Diofantovy priblizheniya v ekstremalnykh zadachakh v $L_2$”, Dokl. AN SSSR, 251:1 (1980), 54–57 | MR | Zbl

[16] V. I. Ivanov, O. I. Smirnov, Konstanty Dzheksona i konstanty Yunga v prostranstvakh $L_p$, Tulskii gos. un-t, Tula, 1995

[17] S. N. Vasilev, “Tochnoe neravenstvo Dzheksona–Stechkina v $L_2$ s modulem nepreryvnosti, porozhdennym proizvolnym konechno-raznostnym operatorom s postoyannymi koeffitsientami”, Dokl. RAN, 385:1 (2002), 11–14 | MR | Zbl

[18] M. G. Esmaganbetov, “Poperechniki klassov iz $L_2[0,2\pi]$ i minimizatsiya tochnykh konstant v neravenstvakh tipa Dzheksona”, Matem. zametki, 65:6 (1999), 816–820 | MR | Zbl

[19] S. B. Vakarchuk, A. N. Schitov, “Nailuchshie polinomialnye priblizheniya v $L_2$ i poperechniki nekotorykh klassov funktsii”, Ukr. matem. zhurn., 56:11 (2004), 1458–1466 | MR | Zbl

[20] A. S. Serdyuk, A. I. Stepanets, “Pryamye i obratnye teoremy teorii priblizhenii funktsii v prostranstve $S^p$”, Ukr. matem. zhurn., 54:1 (2002), 106–124 | MR | Zbl

[21] G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge Univ. Press, Cambrige, 1952 | MR | Zbl

[22] V. M. Tikhomirov, Nekotorye voprosy teorii priblizhenii, Izd-vo Mosk. un-ta, M., 1976 | MR