Geodesic
Sharp estimates for the rate of convergence of double Fourier series in classical orthogonal polynomials
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 7, pp. 1109-1117 
V. A. Abilov; M. V. Abilov; M. K. Kerimov. Sharp estimates for the rate of convergence of double Fourier series in classical orthogonal polynomials. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 7, pp. 1109-1117. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_7_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Sharp estimates are obtained for the convergence rate of “triangular” and “hyperbolic” partial sums of Fourier series in orthogonal (Laguerre, Hermite, Jacobi) polynomials in the classes of differentiable functions of two variables characterized by a generalized modulus of continuity. The proofs are based on the generalized shift operator and generalized modulus of continuity for functions from $\mathbb{L}_2$ having generalized partial derivatives in Levi’s sense.

[1] Suetin P. K., Klassicheskie ortogonalnye mnogochleny, Nauka, M., 1979, 415 pp. | MR

[2] Nikiforov A. F., Uvarov V. B., Spetsialnye funktsii matematicheskoi fiziki, Nauka, M., 1978, 320 pp. | MR

[3] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, Perev. s angl. N. Ya. Vilenkina, v. 2, Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonalnye mnogochleny, Nauka, M., 1966, 266 pp. | MR

[4] Rafalson S. Z., “Nailuchshee priblizhenie funktsii v metrikakh $\mathbb{L}^2_{p(x)}$ algebraicheskimi mnogochlenami i koeffitsienty Fure po ortogonalnym mnogochlenam”, Vestn. LGU. Mekhan. i matem., 1969, no. 7, 68–79 | MR

[5] Abilova F. V., “O nailuchshem priblizhenii funktsii algebraicheskimi mnogochlenami v srednem”, Dokl. na B'lgarskoi AN, 46:12 (1993), 9–11 | MR

[6] Abilov V. A., Abilova F. V., “O nailuchshem priblizhenii funktsii algebraicheskimi mnogochlenami v srednem”, Izvestiya VUZOV. Matem., 1997, no. 3, 40–43 | MR

[7] Abilov V. A., Abilova F. V., Kerimov M. K., “Tochnye otsenki skorosti skhodimosti ryadov Fure na nekotorykh klassakh funktsii v prostranstve $\mathbb{L}_2((a,b),p(x))$”, Zh. vychisl. matem. i matem. fiz., 49:6 (2009), 966–980 | MR | Zbl

[8] Abilov V. A., Kerimov M. K., “Tochnye otsenki skorosti skhodimosti dvoinykh ryadov Fure po ortogonalnym mnogochlenam v prostranstve $\mathbb{L}_2((a,b)\times(c,d); p(x)q(y))$”, Zh. vychisl. matem. i matem. fiz., 49:8 (2009), 1364–1368 | MR | Zbl

[9] Abilov V. A., Kerimov M. K., “Tochnye otsenki skorosti skhodimosti “giperbolicheskikh” chastnykh summ dvoinogo ryada Fure po ortogonalnym mnogochlenam”, Zh. vychisl. matem. i matem. fiz., 52:11 (2012), 1052–1058

[10] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1981, 542 pp. | MR

[11] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969, 480 pp. | MR