Geodesic
Existence of Continuous Functions with a Given Order of Decrease of Least Deviations from Rational Approximations
Matematičeskie zametki, Tome 74 (2003) no. 5, pp. 745-751 
A. P. Starovoitov. Existence of Continuous Functions with a Given Order of Decrease of Least Deviations from Rational Approximations. Matematičeskie zametki, Tome 74 (2003) no. 5, pp. 745-751. http://geodesic.mathdoc.fr/item/MZM_2003_74_5_a10/
@article{MZM_2003_74_5_a10,
     author = {A. P. Starovoitov},
     title = {Existence of {Continuous} {Functions} with a {Given} {Order} of {Decrease} of {Least} {Deviations} from {Rational} {Approximations}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {745--751},
     year = {2003},
     volume = {74},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_5_a10/}
}
TY  - JOUR
AU  - A. P. Starovoitov
TI  - Existence of Continuous Functions with a Given Order of Decrease of Least Deviations from Rational Approximations
JO  - Matematičeskie zametki
PY  - 2003
SP  - 745
EP  - 751
VL  - 74
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/MZM_2003_74_5_a10/
LA  - ru
ID  - MZM_2003_74_5_a10
ER  - 
%0 Journal Article
%A A. P. Starovoitov
%T Existence of Continuous Functions with a Given Order of Decrease of Least Deviations from Rational Approximations
%J Matematičeskie zametki
%D 2003
%P 745-751
%V 74
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_2003_74_5_a10/
%G ru
%F MZM_2003_74_5_a10

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

For a given strictly decreasing sequence $\{a_n\}^\infty_{n=0}$ of real numbers convergent to zero, we construct a continuous function $g$ on the closed interval $[-1,1]$ such that $R_{2n}(g)$ and $a_n$ have identical order of decrease as $n\to\infty$. Here $R_{n}(g)$ are the best approximations on the closed interval $[-1,1]$ in the uniform norm of the function $g$ by algebraic rational functions of degree at most $n$.

[1] Dolzhenko E. P., “Sravnenie skorostei ratsionalnoi i polinomialnoi approksimatsii”, Matem. zametki, 1:3 (1967), 313–320 | Zbl

[2] Bernshtein S. N., “Sur le probléme inverse de la théorie de la meilleure approximation des fonctions continues”, C. R. Acad. Sci., 206 (1938), 1520–1523 | Zbl

[3] Natanson I. P., Konstruktivnaya teoriya funktsii, GITTL, M.–L., 1949

[4] Bernshtein S. N., Sobranie sochinenii v 4-kh tomakh, T. 2, Izd-vo AN SSSR, M., 1954

[5] Bernshtein S. N., Ekstremalnye svoistva polinomov i nailuchshee priblizhenie nepreryvnykh funktsii odnoi veschestvennoi peremennoi, ONTI, M.–L., 1937

[6] Gonchar A. A., “O nailuchshikh priblizheniyakh ratsionalnykh funktsii”, Dokl. AN SSSR, 100:2 (1955), 13–16

[7] Gonchar A. A., “Otsenki rosta ratsionalnykh funktsii i nekotorye ikh prilozheniya”, Matem. sb., 72(114):3 (1967), 489–503 | Zbl

[8] Starovoitov A. P., “K probleme opisaniya posledovatelnostei nailuchshikh trigonometricheskikh ratsionalnykh priblizhenii”, Matem. zametki, 69:6 (2001), 919–924 | MR | Zbl

[9] Starovoitov A. P., “K probleme opisaniya posledovatelnostei nailuchshikh trigonometricheskikh ratsionalnykh priblizhenii”, Matem. sb., 191:6 (2000), 145–154 | MR | Zbl

[10] Pekarskii A. A., “Suschestvovanie funktsii s zadannymi nailuchshimi ravnomernymi ratsionalnymi priblizheniyami”, Izv. AN Belarusi. Ser. fiz.-matem. nauk, 1994, no. 1, 23–26 | Zbl

[11] Rusak V. N., Ratsionalnye funktsii kak apparat priblizheniya, BGU, Minsk, 1979 | MR

[12] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR

[13] Dolzhenko E. P., “Skorost priblizheniya ratsionalnymi drobyami i svoistva funktsii”, Matem. sb., 56(98):4 (1962), 403–432 | MR | Zbl

[14] Lorentz G., v. Golitschek M., Makavoz Y., Constructive Approximation. Advanced Problems, Springer-Verlag, New York–Berlin–Heidelberg, 1996 | MR