Geodesic
Direct and Inverse Theorems on the Approximation of Functions by Fourier–Laplace Sums in the Spaces $S^{(p,q)}(\sigma^{m-1})$
Matematičeskie zametki, Tome 98 (2015) no. 4, pp. 530-543 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, we prove direct and inverse theorems on the approximation of functions by Fourier–Laplace sums in the spaces $S^{(p,q)}(\sigma^{m-1})$, $m\ge 3$, in terms of best approximations and moduli of continuity and consider the constructive characteristics of function classes defined by the moduli of continuity of their elements. The given statements generalize the results of the author's work carried out in 2007.
Keywords: approximation of functions, the spaces $S^{(p,q)}(\sigma^{m-1})$, modulus of continuity, Parseval's equality, Jackson-type inequality, Bernstein–Stechkin–Timan-type inequality.
Mots-clés : Fourier–Laplace sum, Gegenbauer polynomial 
@article{MZM_2015_98_4_a3,
     author = {R. A. Lasuriya},
     title = {Direct and {Inverse} {Theorems} on the {Approximation} of {Functions} by {Fourier{\textendash}Laplace} {Sums} in the {Spaces} $S^{(p,q)}(\sigma^{m-1})$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {530--543},
     year = {2015},
     volume = {98},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2015_98_4_a3/}
}
TY  - JOUR
AU  - R. A. Lasuriya
TI  - Direct and Inverse Theorems on the Approximation of Functions by Fourier–Laplace Sums in the Spaces $S^{(p,q)}(\sigma^{m-1})$
JO  - Matematičeskie zametki
PY  - 2015
SP  - 530
EP  - 543
VL  - 98
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/MZM_2015_98_4_a3/
LA  - ru
ID  - MZM_2015_98_4_a3
ER  - 
%0 Journal Article
%A R. A. Lasuriya
%T Direct and Inverse Theorems on the Approximation of Functions by Fourier–Laplace Sums in the Spaces $S^{(p,q)}(\sigma^{m-1})$
%J Matematičeskie zametki
%D 2015
%P 530-543
%V 98
%N 4
%U http://geodesic.mathdoc.fr/item/MZM_2015_98_4_a3/
%G ru
%F MZM_2015_98_4_a3
R. A. Lasuriya. Direct and Inverse Theorems on the Approximation of Functions by Fourier–Laplace Sums in the Spaces $S^{(p,q)}(\sigma^{m-1})$. Matematičeskie zametki, Tome 98 (2015) no. 4, pp. 530-543. http://geodesic.mathdoc.fr/item/MZM_2015_98_4_a3/

[1] R. A. Lasuriya, “Pryamye i obratnye teoremy priblizheniya funktsii, zadannykh na sfere v prostranstve $S^{(p,q)}(\sigma^m)$”, Ukr. matem. zhurn., 59:7 (2007), 901–911 | MR | Zbl

[2] A. I. Stepanets, Metody teorii priblizhenii, Ch. 2, Pratsi In-tu matem. NAN Ukraïni, 40, In-t matem. NAN Ukraïni, Kiïv, 2002

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

[4] H. Berens, P. L. Butzer, S. Pawelke, “Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten”, Publ. Res. Inst. Math. Sci. Ser. A, 4:2 (1968), 201–268 | DOI | MR | Zbl

[5] S. M. Nikolskii, P. I. Lizorkin, “Approksimatsiya funktsii na sfere”, Izv. AN SSSR. Ser. matem., 51:3 (1987), 635–651 | MR | Zbl

[6] Ar. S. Dzhafarov, “O sfericheskikh analogakh klassicheskikh teorem Dzh. Dzheksona i S. N. Bernshteina”, Dokl. AN SSSR, 203:2 (1972), 278–281 | Zbl

[7] G. G. Kushnirenko, “O priblizhenii funktsii, zadannykh na edinichnoi sfere, konechnymi sfericheskimi summami”, Nauchnye doklady vysshei shkoly. Fiz.-matem. nauki, 4 (1958), 47–53 | Zbl

[8] I. V. Petrova, “Teorema Dzheksona i prostranstvo Besova na sfere”, Dokl. AN SSSR, 278:3 (1984), 544–549 | MR | Zbl

[9] Kh. P. Rustamov, “O priblizhenii funktsii na sfere”, Izv. RAN. Ser. matem., 57:5 (1993), 127–148 | MR | Zbl

[10] P. L. Butzer, “A survey of work on approximation at Aachen, 1968–1972”, Approximation Theory, Academic Press, New York, 1973, 31–100 | MR | Zbl

[11] V. V. Shalaev, “Tochnye otsenki priblizheniya nepreryvnykh na sfere funktsii lineinymi operatorami tipa svertki”, Ukr. matem zhurn., 43:4 (1991), 565–567 | MR | Zbl

[12] V. V. Arestov, V. Yu. Popov, “Neravenstva Dzheksona na sfere v $L_2$”, Izv. vuzov. Matem., 1995, no. 8, 13–20 | MR | Zbl

[13] A. G. Babenko, “Tochnoe neravenstvo Dzheksona–Stechkina v prostranstve $L^2$ funktsii na mnogomernoi sfere”, Matem. zametki, 60:3 (1996), 333–355 | DOI | MR | Zbl

[14] D. V. Gorbachev, “Tochnoe neravenstvo Dzheksona v prostranstve $L_p$ na sfere”, Matem. zametki, 66:1 (1999), 50–62 | DOI | MR | Zbl

[15] V. F. Babenko, V. G. Doronin, A. A. Ligun, A. A. Shumeiko, “O neravenstvakh tipa Dzheksona dlya funktsii, zadannykh na sfere”, Ukr. matem. zhurn., 57:3 (2005), 291–304 | MR | Zbl

[16] V. Yu. Popov, “Priblizhenie na sfere v $L_2$”, Dokl. AN SSSR, 301:4 (1988), 793–797 | MR | Zbl

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

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