Geodesic
The exact order of the best approximation to convex functions by rational functions
Sbornik. Mathematics, Tome 32 (1977) no. 2, pp. 245-251 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We show that the least uniform rational deviations $R_n(f)$ from the function $f(x)$, continuous and convex on the interval $[a,b]$, satisfy the condition $R_n(f)=o(1/n)$ as $n\to\infty$, and that $R_n(f)=O(1/n)$ uniformly for the continuous convex functions $f$ whose absolute values are bounded by unity. These estimates are precise with respect to the rate of decrease of the right-hand sides. Bibliography: 16 titles. 
@article{SM_1977_32_2_a4,
     author = {V. A. Popov and P. P. Petrushev},
     title = {The exact order of the best approximation to convex functions by~rational functions},
     journal = {Sbornik. Mathematics},
     pages = {245--251},
     year = {1977},
     volume = {32},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1977_32_2_a4/}
}
TY  - JOUR
AU  - V. A. Popov
AU  - P. P. Petrushev
TI  - The exact order of the best approximation to convex functions by rational functions
JO  - Sbornik. Mathematics
PY  - 1977
SP  - 245
EP  - 251
VL  - 32
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1977_32_2_a4/
LA  - en
ID  - SM_1977_32_2_a4
ER  - 
%0 Journal Article
%A V. A. Popov
%A P. P. Petrushev
%T The exact order of the best approximation to convex functions by rational functions
%J Sbornik. Mathematics
%D 1977
%P 245-251
%V 32
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1977_32_2_a4/
%G en
%F SM_1977_32_2_a4
V. A. Popov; P. P. Petrushev. The exact order of the best approximation to convex functions by rational functions. Sbornik. Mathematics, Tome 32 (1977) no. 2, pp. 245-251. http://geodesic.mathdoc.fr/item/SM_1977_32_2_a4/

[1] D. Newman, “Rational approximation to $|x|$”, Michigan Math. J., 11 (1964), 11–14 | DOI | MR | Zbl

[2] P. Szusz, P. Turan, “On the constructive theory of function, I”, Publ. Math. Inst. Hung. Acad. Sci., 9 (1965), 495–502 | MR

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

[4] A. A. Gonchar, “O skorosti ratsionalnoi approksimatsii nepreryvnykh funktsii s kharakternymi osobennostyami”, Matem. sb., 73 (115) (1967), 630–638 | MR | Zbl

[5] G. Freud, “Über die Approximation reeler Funktionen durch rationale gebrochene Funktionen”, Acta Math. Acad. Sci. Hung., 17 (1966), 313–324 | DOI | MR | Zbl

[6] A. P. Bulanov, “O poryadke priblizheniya vypuklykh funktsii ratsionalnymi funktsiyami”, Izv. AN SSSR, seriya matem., 33 (1969), 1132–1148 | MR | Zbl

[7] A. P. Bulanov, “Ratsionalnye priblizheniya vypuklykh funktsii s zadannym modulem nepreryvnosti”, Matem. sb., 84 (126) (1971), 476–494 | MR | Zbl

[8] J. Szabados, “Rational approximation of analytic functions with finite number of singularities on the real axis”, Acta Math. Acad. Sci. Hung., 20 (1969), 159–167 | DOI | MR | Zbl

[9] V. A. Popov, J. Szabados, “On a general localization theorem and some applications in the theory of rational approximation”, Acta Math. Acad. Sci. Hung., 25 (1974), 165–170 | DOI | MR | Zbl

[10] A. A. Abdugapparov, “O ratsionalnykh priblizheniyakh funktsii s vypukloi proizvodnoi”, DAN Uz. SSR, 10 (1972), 3–4 | MR | Zbl

[11] V. A. Popov, “Rational uniform approximation of class $V_r$ and its applications”, C. r. de l'Acad. Bulg. des scient., 29:6 (1976), 791–794 | MR | Zbl

[12] P. P. Petrushev, “O ratsionalnoi approksimatsii funktsii s vypukloi proizvodnoi”, DAN Bolgarii, 29:10 (1976)

[13] A. Khatamov, “O ratsionalnom priblizhenii vypuklykh funktsii klassa $\mathrm{Lip}\,\alpha$”, Matem. zametki, 18:6 (1975), 845–854 | MR | Zbl

[14] V. A. Popov, “Ob approksimatsii absolyutno nepreryvnykh funktsii splain-funktsiyami”, DAN Bolgarii, 28:10 (1975), 1299–1301 | MR | Zbl

[15] A. A. Abdugapparov, “O ratsionalnykh priblizheniyakh vypuklykh funktsii”, Izv. AN Uz. SSSR, seriya fiz.-matem., 1969, no. 2, 3–9 | MR | Zbl

[16] A. A. Abdugapparov, “O ratsionalnykh priblizheniyakh vypuklykh funktsii klassa $\mathrm{Lip}\,\alpha$”, Izv. AN Uz. SSR, seriya fiz.-matem., 1972, no. 6, 5–10 | MR | Zbl