Geodesic
Approximation, by rational functions, of convex functions with given modulus of continuity
Sbornik. Mathematics, Tome 34 (1978) no. 1, pp. 1-24 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We denote by $R_n[f]$ the least deviation of the continuous function $f(x)$, $x\in[a,b]$, from the rational functions of order at most $n$. We establish the following theorems. Theorem 1. Let $f(x)$ be convex on $[a,b]$ $(-\infty with modulus of continuity $\omega(\delta,f)$. Then $$ R_n[f]\leqslant c\frac{\ln^6n}{n^2}\max_{(b-a)e^{-n}\leqslant\theta\leqslant b-a}\biggl\{\omega(\theta)\ln\frac{b-a}{\theta}\biggr\},\qquad n=2,3,\dots, $$ where $c$ is an absolute constant. \medskip Theorem 2. There exist a convex function $f^*(x)$ and a sequence $n_k\nearrow\infty$ such that 1) $\omega(\delta,f^*)\leqslant(\ln(e/\delta))^{-\gamma}$, $0<\delta\leqslant1$, and 2) $R_{n_k}[f^*]\geqslant c_1\gamma/n^{1-\gamma}_k$, where $c_1$ is an absolute constant. Bibliography: 8 titles. 
@article{SM_1978_34_1_a0,
     author = {A. P. Bulanov},
     title = {Approximation, by rational functions, of convex functions with given modulus of continuity},
     journal = {Sbornik. Mathematics},
     pages = {1--24},
     year = {1978},
     volume = {34},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1978_34_1_a0/}
}
TY  - JOUR
AU  - A. P. Bulanov
TI  - Approximation, by rational functions, of convex functions with given modulus of continuity
JO  - Sbornik. Mathematics
PY  - 1978
SP  - 1
EP  - 24
VL  - 34
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1978_34_1_a0/
LA  - en
ID  - SM_1978_34_1_a0
ER  - 
%0 Journal Article
%A A. P. Bulanov
%T Approximation, by rational functions, of convex functions with given modulus of continuity
%J Sbornik. Mathematics
%D 1978
%P 1-24
%V 34
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1978_34_1_a0/
%G en
%F SM_1978_34_1_a0
A. P. Bulanov. Approximation, by rational functions, of convex functions with given modulus of continuity. Sbornik. Mathematics, Tome 34 (1978) no. 1, pp. 1-24. http://geodesic.mathdoc.fr/item/SM_1978_34_1_a0/

[1] P. Szüsz, P. Turan, “On the constructive theory of function”, Magyar tud. Acad. Mat. Kutato inter, közl. A, 9:3 (1965), 495–502 | MR

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

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

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

[5] V. A. Popov, “On the rational approximation of functions of the class Vr”, Acta Math. Acad. Sci. Hung., 25:1–2 (1974), 61–65 | DOI | MR | Zbl

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

[7] A. Khatamov, “O ratsionalnykh priblizheniyakh vypuklykh funktsii”, Matem, zametki, 21:3 (1977), 355–370 | Zbl

[8] N. I. Akhiezer, Lektsii po teorii approksimatsii, izd-vo “Nauka”, Moskva, 1965 | MR