Geodesic
Approximation in the mean of classes of differentiable functions by algebraic polynomials
Izvestiya. Mathematics, Tome 23 (1984) no. 2, pp. 353-365 
V. A. Kofanov. Approximation in the mean of classes of differentiable functions by algebraic polynomials. Izvestiya. Mathematics, Tome 23 (1984) no. 2, pp. 353-365. http://geodesic.mathdoc.fr/item/IM2_1984_23_2_a5/
@article{IM2_1984_23_2_a5,
     author = {V. A. Kofanov},
     title = {Approximation in the mean of classes of differentiable functions by algebraic polynomials},
     journal = {Izvestiya. Mathematics},
     pages = {353--365},
     year = {1984},
     volume = {23},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1984_23_2_a5/}
}
TY  - JOUR
AU  - V. A. Kofanov
TI  - Approximation in the mean of classes of differentiable functions by algebraic polynomials
JO  - Izvestiya. Mathematics
PY  - 1984
SP  - 353
EP  - 365
VL  - 23
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/IM2_1984_23_2_a5/
LA  - en
ID  - IM2_1984_23_2_a5
ER  - 
%0 Journal Article
%A V. A. Kofanov
%T Approximation in the mean of classes of differentiable functions by algebraic polynomials
%J Izvestiya. Mathematics
%D 1984
%P 353-365
%V 23
%N 2
%U http://geodesic.mathdoc.fr/item/IM2_1984_23_2_a5/
%G en
%F IM2_1984_23_2_a5

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

The exact values $E_n(W^r_L)_L$ are found for the best approximations in the mean of the function classes $$W^r_L=\{f:f^{(r-1)}\text{ is absolutely continuous, }\|f^{(r)}\|_L\leqslant1\},\qquad r =2,3,\dots,$$ by algebraic polynomials of degree at most $n$ on the interval $[-1,1]$. It is proved that $E_n(W^r_L)_L$ coincides with the uniform norm of the perfect spline $$ \frac1{r!}\biggl[(x+1)^r+2\sum^{n+1}_{i=1}(-1)^i(x-x_i)^r_+\biggr] $$ with nodes $x_i=-\cos\frac{i\pi}{n+2}$. Bibliography: 6 titles.

[1] Nikolskii S. M., “O nailuchshem lineinom metode priblizheniya mnogochlenami v srednem differentsiruemykh funktsii”, Dokl. AN SSSR, 58:2 (1947), 185–188

[2] Kofanov V. A., “Tochnye znacheniya nailuchshikh priblizhenii algebraicheskimi mnogochlenami v srednem klassov $W_L^r$ ($r=1,2$)”, Issledovaniya po sovremennym problemam summirovaniya i priblizheniya funktsii i ikh prilozheniya, DGU, Dnepropetrovsk, 1977, 18–20

[3] Daugavet I. K., Vvedenie v teoriyu priblizheniya funktsii, LGU, L., 1977 | MR

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

[5] Boyanov B. D., “Nailuchshie metody interpolirovaniya dlya nekotorykh klassov differentsiruemykh funktsii”, Matem. zametki, 17:4 (1975), 511–524 | MR | Zbl

[6] Nikolskii S. M., “O nailuchshem priblizhenii mnogochlenami v srednem funktsii $|a-x|^s$”, Izv. AN SSSR. Ser. matem., 11 (1947), 139–180