Geodesic
Approximation Properties of the Operators $\mathscr Y_{n+2r}(f)$ and of Their Discrete Analogs
Matematičeskie zametki, Tome 72 (2002) no. 5, pp. 765-795.

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper is devoted to the study of the approximation properties of linear operators which are partial Fourier–Legendre sums of order $n$ with $2r$ terms of the form $\sum _{k=1}^{2r}a_kP_{n+k}(x)$ added; here $P_m(x)$ denotes the Legendre polynomial. Due to this addition, the linear operators interpolate functions and their derivatives at the endpoints of the closed interval $[-1,1]$, which, in fact, for $r=1$ allows us to significantly improve the approximation properties of partial Fourier–Legendre sums. It is proved that these operators realize order-best uniform algebraic approximation of the classes of functions $W_rH_{L_2}^\mu $ and $A_q(B)$. With the aim of the computational realization of these operators, we construct their discrete analogs by means of Chebyshev polynomials, orthogonal on a uniform grid, also possessing nice approximation properties. 
@article{MZM_2002_72_5_a15,
     author = {I. I. Sharapudinov},
     title = {Approximation {Properties} of the {Operators} $\mathscr Y_{n+2r}(f)$ and of {Their} {Discrete} {Analogs}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {765--795},
     publisher = {mathdoc},
     volume = {72},
     number = {5},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_72_5_a15/}
}
TY  - JOUR
AU  - I. I. Sharapudinov
TI  - Approximation Properties of the Operators $\mathscr Y_{n+2r}(f)$ and of Their Discrete Analogs
JO  - Matematičeskie zametki
PY  - 2002
SP  - 765
EP  - 795
VL  - 72
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2002_72_5_a15/
LA  - ru
ID  - MZM_2002_72_5_a15
ER  - 
%0 Journal Article
%A I. I. Sharapudinov
%T Approximation Properties of the Operators $\mathscr Y_{n+2r}(f)$ and of Their Discrete Analogs
%J Matematičeskie zametki
%D 2002
%P 765-795
%V 72
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2002_72_5_a15/
%G ru
%F MZM_2002_72_5_a15
I. I. Sharapudinov. Approximation Properties of the Operators $\mathscr Y_{n+2r}(f)$ and of Their Discrete Analogs. Matematičeskie zametki, Tome 72 (2002) no. 5, pp. 765-795. http://geodesic.mathdoc.fr/item/MZM_2002_72_5_a15/

[1] Sharapudinov I. I., “Priblizhenie funktsii s peremennoi gladkostyu summami Fure–Lezhandra”, Matem. sb., 191:5 (2000), 143–160 | MR | Zbl

[2] Sharapudinov I. I., “Priblizhenie diskretnykh funktsii i mnogochleny Chebysheva, ortogonalnye na ravnomernoi setke”, Matem. zametki, 67:3 (2000), 460–470 | MR | Zbl

[3] Sharapudinov I. I., Mnogochleny, ortogonalnye na setkakh. Teoriya i prilozheniya, Izd-vo Dag. gos. ped. un-ta, Makhachkala, 1997

[4] Segë G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962

[5] Gasper G., “Positivity and special functions”, Theory and Appl. Spec. Funct., ed. R. A. Askey, 1975, 375–433 | MR | Zbl

[6] Timan A. F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960

[7] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1965

[8] Karlin S., McGregor J. L., “The Hahn polynomials, formulas, and an application”, Scripta Math., 26 (1961), 33–46 | MR | Zbl