Geodesic
Approximation of Classes of Convolutions by Linear Operators of Special Form
Matematičeskie zametki, Tome 90 (2011) no. 3, pp. 351-361 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A parametric family of operators $G_\rho$ is constructed for the class of convolutions $\mathbf{W}_{p,m}(K)$ whose kernel $K$ was generated by the moment sequence. We obtain a formula for evaluating $$ E(\mathbf{W}_{p,m}(K);G_\rho)_p:=\sup_{f\in\mathbf{W}_{p,m}(K)}\|f-G_\rho(f)\|_p. $$ For the case in which $\mathbf{W}_{p,m}(K)=\mathbf{W}^{r,\beta}_{p,m}$, we obtain an expansion in powers of the parameter $\varepsilon=-\ln\rho$ for $E(\mathbf{W}^{r,\beta}_{p,m};G_{\rho,r})_p$, where $\beta\in\mathbb{Z}$, $r>0$, and $m\in\mathbb{N}$, while $p=1$ or $p=\infty$.
Mots-clés : convolution, Euler polynomial
Keywords: linear operator, periodic measurable function, moment sequence, Borel measure, Fourier series, Bernoulli numbers. 
@article{MZM_2011_90_3_a2,
     author = {V. P. Zastavnyi and V. V. Savchuk},
     title = {Approximation of {Classes} of {Convolutions} by {Linear} {Operators} of {Special} {Form}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {351--361},
     year = {2011},
     volume = {90},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_90_3_a2/}
}
TY  - JOUR
AU  - V. P. Zastavnyi
AU  - V. V. Savchuk
TI  - Approximation of Classes of Convolutions by Linear Operators of Special Form
JO  - Matematičeskie zametki
PY  - 2011
SP  - 351
EP  - 361
VL  - 90
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/MZM_2011_90_3_a2/
LA  - ru
ID  - MZM_2011_90_3_a2
ER  - 
%0 Journal Article
%A V. P. Zastavnyi
%A V. V. Savchuk
%T Approximation of Classes of Convolutions by Linear Operators of Special Form
%J Matematičeskie zametki
%D 2011
%P 351-361
%V 90
%N 3
%U http://geodesic.mathdoc.fr/item/MZM_2011_90_3_a2/
%G ru
%F MZM_2011_90_3_a2
V. P. Zastavnyi; V. V. Savchuk. Approximation of Classes of Convolutions by Linear Operators of Special Form. Matematičeskie zametki, Tome 90 (2011) no. 3, pp. 351-361. http://geodesic.mathdoc.fr/item/MZM_2011_90_3_a2/

[1] N. P. Korneichuk, Ekstremalnye zadachi teorii priblizheniya, Nauka, M., 1976 | MR

[2] S. M. Nikolskii, “Priblizhenie funktsii trigonometricheskimi polinomami v srednem”, Izv. AN SSSR. Ser. matem., 10:3 (1946), 207–256 | MR | Zbl

[3] V. V. Zhuk, Approksimatsiya periodicheskikh funktsii, Izd-vo LGU, L., 1982 | MR | Zbl

[4] S. A. Telyakovskii, “O priblizhenii funktsii, udovletvoryayuschikh usloviyu Lipshitsa, summami Feiera”, Ukr. matem. zhurn., 21:3 (1969), 334–343 | MR | Zbl

[5] V. A. Baskakov, S. A. Telyakovskii, “O priblizhenii differentsiruemykh funktsii summami Feiera”, Matem. zametki, 32:2 (1982), 129–140 | MR | Zbl

[6] V. P. Zastavnyi, “O ryadakh, voznikayuschikh pri priblizhenii periodicheskikh differentsiruemykh funktsii integralami Puassona”, Matem. zametki, 86:4 (2009), 497–511 | MR | Zbl

[7] A. Zigmund, Trigonometricheskie ryady, v. 1, Mir, M., 1965 | MR | Zbl

[8] R. Edvards, Ryady Fure v sovremennom izlozhenii, v. 1, Mir, M., 1985 | MR | Zbl

[9] V. V. Savchuk, “Priblizhenie $2\pi$-periodicheskikh i golomorfnykh funktsii nekotorymi lineinymi metodami”, Sb. trudov In-ta matem. NAN Ukrainy, 5:1 (2008), 309–323 | Zbl

[10] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1973 | MR | Zbl