On the best polynomial approximation in Hardy space

M. Sh. Shabozov ; Z. Sh. Malakbozov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2022), pp. 110-123.

Résumé

Sharp Jackson-Stechkin-type inequalities in which the best polynomial approximation of a function in the Hardy space $H_2$ is estimated from above both in terms of the generalized modulus of continuity of the $m$-th order and in terms of the $\mathcal{K}$-functional of $r$-th derivatives are found. For some classes of functions defined with the formulated characteristics in the space $H_2$, the exact values of $n$-widths are calculated. Also in the classes $W_{2}^{(r)}(\widetilde{\omega}_{m},\Phi)$ and $W_{2}^{(r)}(\mathcal{K}_{m},\Phi)$, where $r\in\mathbb{N}$, $r\ge2$ the exact values of the best polynomial approximations of intermediate derivatives $f^{(s)}$, $1\le s\le r-1$ are obtained.

Détail

Keywords: the best polynomial approximation, generalized modulus of continuity, $\mathcal{K}$-functional, characteristic of smoothness, $n$-width.

Bibliographie

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

[2] Taikov L. V., “O nailuchshikh lineinykh metodakh priblizheniya funktsii klassov $\mathcal{B}^{r}$ i $\mathcal{H}^{r}$”, UMN, 18(112):4 (1963), 183–189 | MR

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

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

[5] 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

[6] Taikov L. V., “Nekotorye tochnye neravenstva v teorii priblizheniya funktsii”, Anal. Math., 2:1 (1976), 77–85 | DOI | MR

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

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

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

[11] Vakarchuk S. B., “O nailuchshikh lineinykh metodakh priblizheniya i poperechnikakh nekotorykh klassov analiticheskikh funktsii”, Matem. zametki, 65:2 (1999), 186–193

[12] Vakarchuk S. B., “Tochnye znacheniya poperechnikov klassov analiticheskikh v kruge funktsii i nailuchshie lineinye metody priblizheniya”, Matem. zametki, 72:5 (2002), 665–669 | MR

[13] Vakarchuk S. B., Zabutnaya V. I., “O nailuchshikh lineinykh metodakh priblizheniya funktsii klassov L. V.Taikova v prostranstvakh Khardi $H_{q,\rho},~q\ge1,$ $0\rho\le1$”, Matem. zametki, 85:3 (2009), 323–329 | MR

[14] Shabozov M. Sh., Shabozov O. Sh., “Poperechniki nekotorykh klassov analiticheskikh funktsii v prostranstve Khardi $H_2$”, Matem. zametki, 68:5 (2000), 796–800 | MR

[15] Shabozov M. Sh., Yusupov G. A., “Nailuchshee priblizhenie i znacheniya poperechnikov nekotorykh klassov analiticheskikh funktsii”, DAN Rossii, 382:6 (2002), 747–749 | MR

[16] Shabozov M. Sh., Pirov Kh. Kh., “Tochnye konstanty v neravenstvakh tipa Dzheksona dlya priblizheniya analiticheskikh funktsii iz $H^{R}_p$, $1\le p\le2$”, DAN Rossii, 394:3 (2004), 317–319 | MR

[17] Shabozov M. Sh., Langarshoev M. R., “O nailuchshikh lineinykh metodakh priblizheniya nekotorykh klassov analiticheskikh v edinichnom kruge funktsii”, Sib. matem. zhurn., 60:6 (2019), 1414–1423 | MR

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

[19] Vakarchuk S. B., Vakarchuk M. 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

[20] Vakarchuk S. B., “Obobschennye kharakteristiki gladkosti v neravenstvakh tipa Dzheksona i poperechniki klassov funktsii v $L_{2}$”, Matem. zametki, 98:4 (2015), 511–529

[21] Shabozov M. Sh., Shabozova A. A., “Nekotorye tochnye neravenstva tipa Dzheksona–Stechkina dlya periodicheskikh differentsiruemykh v smysle Veilya funktsii v $L_2$”, Tr. IMM UrO RAN, 25, no. 4, 2019, 255–264

[22] 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

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

[24] Vakarchuk S. B., “Priblizhenie funktsii v srednem na veschestvennoi osi algebraicheskimi polinomami s vesom Chebysheva–Ermita i poperechniki funktsionalnykh klassov”, Matem. zametki, 95:5 (2014), 666–684

[25] 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

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

[27] Pinkus A., $n$-Widths in Approximation Theory, Springer-Verlag, New York–Tokyo–Berlin–Heidelberg, 1985 | MR

[28] Shevchuk I. A., Priblizhenie mnogochlenami i sledy nepreryvnykh na otrezke funktsii, Naukova dumka, Kiev, 1992