On simultaneous approximation of certain classes of functions in the Bergman space $B_2$

M. Sh. Shabozov ; A. A. Shabozova

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2024), pp. 80-88

Résumé

The problem of finding the supremums of the best simultaneous polynomial approximations of some classes of functions analytic in the unit disk and belonging to the Bergman space $B_2$ is considered. The indicated function classes are defined by the averaged values of the $m$th order moduli of continuity of the highest derivative bounded from above by some majorant $\Phi$.

Détail

Keywords: simultaneous approximation, upper bound, best approximation, Bergman space, modulus of continuity
Mots-clés : majorant.

Bibliographie

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

[2] Tikhomirov V.M., Nekotorye voprosy teorii priblizhenii, MGU, M., 1976

[3] Babenko K.I., “O nailuchshikh priblizheniyakh odnogo klassa analiticheskikh funktsii”, Izv. AN SSSR. Ser. matem., 22:5 (1958), 631–640 | Zbl

[4] Tikhomirov V.M., “Teoriya priblizhenii”, Analiz–2, Itogi nauki i tekhn. Ser. Sovrem. probl. matem. Fundam. napravleniya, 14, VINITI, M., 1987, 103–260

[5] Taikov L.V., “O nailuchshem priblizhenii v srednem nekotorykh klassov analiticheskikh funktsii”, Matem. zametki, 1:2 (1967), 155–162 | MR | Zbl

[6] Taikov L.V., “Poperechniki nekotorykh klassov analiticheskikh funktsii”, Matem. zametki, 22:2 (1977), 285–295 | MR | Zbl

[7] Dveirin M.Z., Chebanenko I.V., “O polinomialnoi approksimatsii v banakhovykh prostranstvakh analiticheskikh funktsii”, Teoriya otobrazhenii i priblizhenie funktsii, Naukova dumka, Kiev, 1983

[8] Ainulloev N, Taikov L.V., “Nailuchshie priblizheniya v smysle A.N.Kolmogorova klassov analiticheskikh v edinichnom kruge funktsii”, Matem. zametki, 40:3 (1986), 341–351 | MR

[9] Vakarchuk S.B., “O poperechnikakh nekotorykh klassov analiticheskikh v edinichnom kruge funktsii I”, Ukr. matem. zhurnal, 42:7 (1990), 873–881 | MR | Zbl

[10] Vakarchuk S.B., “O poperechnikakh nekotorykh klassov analiticheskikh v edinichnom kruge funktsii II”, Ukr. matem. zhurnal, 42:8 (1990), 1019–1026 | MR | Zbl

[11] Vakarchuk S.B., “Nailuchshie lineinye metody priblizheniya i poperechniki klassov analiticheskikh v kruge funktsii”, Matem. zametki, 57:1 (1995), 30–39 | MR

[12] Vakarchuk S.B., Shabozov M.Sh., “O poperechnikakh klassov funktsii, analiticheskikh v kruge”, Matem. sb., 201:8 (2010), 3–22 | DOI | MR | Zbl

[13] Shabozov M.Sh., Yusupov G.A., “Nailuchshie metody priblizheniya i znacheniya poperechnikov nekotorykh klassov funktsii v prostranstve $H_{q,\rho}\,1\le q\le\infty, 0\rho\le1$”, Sib. matem. zhurn., 57:2 (2016), 469–478 | MR | Zbl

[14] Farkov Yu.A., “O nailuchshem lineinom priblizhenii golomorfnykh funktsii”, Fundament. i prikl. matem., 19:5 (2014), 185–212

[15] Parfenov O.G., “Poperechniki odnogo klassa analiticheskikh funktsii”, Matem. sb., 117(159):2 (1982), 279–285 | MR | Zbl

[16] Osipenko K.Yu., Stesin M.I., “O poperechnikakh klassa Khardi $H_2$ v $n$-mernom share”, UMN, 45:5(275) (1990), 193–194 | MR | Zbl

[17] Vakarchuk S.B., Zabutnaya V.I., “O nailuchshikh lineinykh metodakh priblizheniya funktsii klassov L.V.Taikova v prostranstvakh Khardi $H_{q,\rho}\, q\ge1,\, 0\rho\le 1$”, Matem. zametki, 85:3 (2009), 323–329 | DOI | MR | Zbl

[18] Shabozov M.Sh., Saidusainov M.S., “Srednekvadraticheskoe priblizhenie funktsii kompleksnogo peremennogo summami Fure po ortogonalnym sistemam”, Tr. IMM UrO RAN, 25, no. 2, 2019, 258–272

[19] Scheik J.T., “Polynomial approximation of functions analytic in a disk”, Proc. Amer. Math. Soc., 17:6 (1966), 1238–1243 | DOI | MR

[20] Garkavi A.L., “O sovmestnom priblizhenii periodicheskoi funktsii i ee proizvodnykh trigonometricheskimi polinomami”, Izv. AN SSSR. Ser. matem., 24:1 (1960), 103–128 | Zbl

[21] Timan A.F., “K voprosu ob odnovremennoi approksimatsii funktsii i ikh proizvodnykh na vsei chislovoi osi”, Izv. AN SSSR. Ser. matem., 24:3 (1960), 421–430 | Zbl

[22] Malozemov V.N., Sovmestnoe priblizhenie funktsii i ee proizvodnykh, LGU, L., 1973

[23] Vakarchuk S.B., Zabutnaya V.I., “Neravenstva tipa Dzheksona–Stechkina dlya spetsialnykh modulei nepreryvnosti i poperechniki funktsionalnykh klassov v prostranstve $L_2$”, Matem. zametki, 92:4 (2012), 497–514 | DOI | Zbl

[24] Shabozov M.Sh., Kadamshoev N.U., “Tochnye neravenstva mezhdu nailuchshimi srednekvadraticheskimi priblizheniyami analiticheskikh v kruge funktsii i nekotorymi kharakteristikami gladkosti v prostranstve Bergmana”, Matem. zametki, 110:2 (2021), 266–281 | DOI | MR | Zbl

[25] Shabozov M.Sh., Yusupov G.A., Zargarov D.D., “O nailuchshei sovmestnoi polinomialnoi approksimatsii funktsii i ikh proizvodnykh v prostranstve Khardi”, Tr. IMM UrO RAN, 27, no. 4, 2021, 239–254

[26] Smirnov V.I., Lebedev N.A., Konstruktivnaya teoriya funktsii kompleksnogo peremennogo, Nauka, M.–L., 1964 | MR

[27] Shabozov M.Sh., “O nailuchshem sovmestnom priblizhenii funktsii v prostranstve Bergmana $B_2$”, Matem. zametki, 114:3 (2023), 435–446 | DOI | MR | Zbl

[28] Vakarchuk M.B., Vakarchuk S.B., “Neravenstva tipa Kolmogorova dlya analiticheskikh funktsii odnoi i dvukh kompleksnykh peremennykh i ikh prilozhenie k teorii approksimatsii”, Ukr. matem. zhurn., 63:12 (2011), 1579–1601

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

[30] Ainulloev N., “Znachenie poperechnikov nekotorykh klassov differentsiruemykh funktsii v $L_2$”, DAN Tadzh. SSR, 27:8 (1984), 415–418 | MR