On the best simultaneous “angle” approximation in the mean of periodic functions of two variables from some classes

M. O. Akobirshoev

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2024), pp. 24-36

Résumé

In the $L_2$ metric, we obtain sharp inequalities between the best joint approximations of $2\pi$-periodic functions $f(x,y)$ differentiable in each of the variables and their successive derivatives $f^{(\mu,\nu)}( x,y) \ (\mu=0,1,\ldots,r; \nu=0,1,\ldots,s)$ by trigonometric “angles” with double integrals containing mixed moduli of continuity of higher orders of higher derivatives. The sharp values of the upper bound of the best joint approximation of some classes of functions given by the specified moduli of continuity are found.

Détail

Keywords: the best joint approximation, trigonometric “angle”, mixed modulus of continuity.
Mots-clés : quasi-polynomial

Bibliographie

[1] Potapov M.K., “O priblizhenii «uglom»”, Tr. konf. po konstruktiv. teor. funkts. (Vengriya, Budapesht, 1971), 1971 | Zbl

[2] Potapov M.K., “Priblizhenie «uglom» i teoremy vlozheniya”, Math. Balkanica, 1972, no. 2, 183–198 | MR | Zbl

[3] Tomich M., “O priblizhenii uglom funktsii s dominiruyuschim modulem gladkosti”, Publ. De L'inst. Math. (Beograd), 23:37 (1972), 193–206

[4] Vakarchuk S.B., “O nailuchshem priblizhenii obobschennymi polinomami v odnom prostranstve analiticheskikh funktsii dvukh kompleksnykh peremennykh”, Izv. vuzov. Matem., 1991, no. 7, 14–25 | Zbl

[5] Vakarchuk S.B., Shabozov M.Sh., “O tochnykh znacheniyakh kvazipoperechnikov nekotorykh funktsionalnykh klassov”, Ukr. matem. zhurn., 48:3 (1996), 301–308 | MR

[6] Shabozov M.Sh., Akobirshoev M.O., “Kvazipoperechniki nekotorykh klassov differentsiruemykh periodicheskikh funktsii dvukh peremennykh”, Dokl. RAN, 404:4 (2005), 406–464

[7] Shabozov M.Sh., Akobirshoev M.O., “O tochnykh znacheniyakh kvazipoperechnikov nekotorykh klassov differentsiruemykh periodicheskikh funktsii dvukh peremennykh”, Ukr. mat. zhurn., 61:6 (2009), 855–864 | MR | Zbl

[8] Shabozov M.Sh., Akobirshoev M.O., “Exact estimates of quasiwidths of some classes of differentiable periodic functions of two variables”, Anal. Math., 35:1 (2009), 61–72 | DOI | MR | Zbl

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

[10] Vakarchuk S.B., Shvachko A.V., “O nailuchshem priblizhenii “uglom” v srednem na ploskosti $\mathbb{R}^{2}$ s vesom Chebysheva–Ermita”, Zbirn. prats In-tu matem. NAN Ukraini, 11:3 (2014), 35–46 | Zbl

[11] Vakarchuk S.B., Shvachko A.V., “Neravenstva kolmogorovskogo tipa dlya proizvodnykh funktsii dvukh peremennykh i ikh prilozhenie k approksimatsii “uglom””, Izv. vuzov. Matem., 2015, no. 11, 3–22

[12] Shabozov M.Sh., Akobirshoev M.O., “O neravenstvakh tipa Kolmogorova dlya periodicheskikh funktsii dvukh peremennykh v $L_2$”, Chebyshevsk. sb., 20:2 (2019), 348–365 | DOI | MR | Zbl

[13] Vakarchuk S.B., “Exact constant in an inequality of Jackson type for $L_2$-approximation on the line and exact values of mean widths of functional classes”, East J. Approxim., 10:1–2 (2004), 27–39 | MR | Zbl

[14] Vakarchuk S.B., “Tochnye konstanty v neravenstvakh tipa Dzheksona i tochnye znacheniya poperechnikov funktsionalnykh klassov iz $L_{2}$”, Matem. zametki, 78:5 (2005), 792–796 | DOI | MR | Zbl

[15] Brudnyi Yu.A., “Priblizhenie funktsii $n$ peremennykh kvazimnogochlenami”, Izv. AN SSSR. Ser. Matem., 34:3 (1970), 564–583 | Zbl

[16] Vakarchuk S.B., Zabutnaya V.I., “Tochnoe neravenstvo tipa Dzheksona–Stechkina v $L_{2}$ i poperechniki funktsionalnykh klassov”, Matem. zametki, 86:3 (2009), 328–336 | DOI | MR | Zbl

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