Geodesic
On the dependence of properties of functions on their degree of approximation by polynomials
Izvestiya. Mathematics, Tome 12 (1978) no. 2, pp. 255-288 
E. P. Dolzhenko; E. A. Sevast'yanov. On the dependence of properties of functions on their degree of approximation by polynomials. Izvestiya. Mathematics, Tome 12 (1978) no. 2, pp. 255-288. http://geodesic.mathdoc.fr/item/IM2_1978_12_2_a4/
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     author = {E. P. Dolzhenko and E. A. Sevast'yanov},
     title = {On the dependence of properties of functions on their degree of approximation by polynomials},
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     url = {http://geodesic.mathdoc.fr/item/IM2_1978_12_2_a4/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $f(x)$ be a bounded $2\pi$-periodic function with modulus of continuity $\omega(\delta,f)$; let $E_n(f)$ and $H_\alpha E_n(f)$ be the minimum deviations of $f$ from the trigonometric polynomials of order $\leqslant n$, in the uniform metric and the Hausdorff metric of order $\alpha$, respectively; let $$ \sigma_n(f,\alpha)=H_\alpha E_0(f)+\dots+H_\alpha E_{n-1}(f). $$ Then \begin{gather*} H_\alpha E_n(f)\leqslant E_n(f)\leqslant H_\alpha E_n(f)\exp\{(3+2\sqrt2\,)\alpha\sigma_n(f,\alpha)\},\\ \omega\left(\frac1n,f\right)\leqslant\frac{\exp\{(3+2\sqrt{2})\alpha{\sigma_n}(f,\alpha)\}-1}{n\alpha}. \end{gather*} If $H_\alpha E_n(f)\leqslant c/n\alpha$ as $n\to\infty$, then if $c\pi$ the function $f$ is continuous almost everywhere; if $c\pi/2$ it is continuous everywhere, and if $c1$ we have $f\in\operatorname{Lip}\gamma(c)$, $\gamma(c)>0$. Approximation by algebraic polynomials is also considered, and some corollaries are given. Bibliography: 13 titles.

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[2] Sendov B., “Aproksimirane na funktsii s algebraichni polinomi po otnoshenie na edna metrika ot khausdorfovski tip”, God. na Sof. univ., Fiz.-matem. fak., 55, 1962, 1–39 | MR

[3] Sendov B., “V'rkhu nai-dobroto priblizhenie s algebraichni polinomi po otnoshenie na khausdorfovo razstoyanie”, God. na Sof. univ., Fiz.-matem. fak., 56, 1963, 195–207 | MR

[4] Sendov Bl., Popov V. A., “Analog teoremy S. M. Nikolskogo dlya priblizheniya funktsii algebraicheskimi mnogochlenami v khausdorfovoi metrike”, Trudy Mezhdunarodnoi Konferentsii po konstruktivnoi teorii funktsii, Varna, 1970, 95–105 | MR

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

[6] Salem R., “Sur certaines fonctions continues et les proprietes de leurs series de Fourier”, Comptes Rendus, 201:17 (1935), 703–705 | Zbl

[7] Stechkin S. B., “O poryadke nailuchshikh priblizhenii nepreryvnykh funktsii”, Izv. AN SSSR. Seriya matem., 15 (1951), 219–242 | Zbl

[8] Bernshtein S. N., “O nailuchshem priblizhenii nepreryvnykh funktsii posredstvom mnogochlenov dannoi stepeni”, 1912, Sobranie sochinenii, t. I, AN SSSR, 1952, 11–104

[9] Nikolskii S. M., “O nailuchshem priblizhenii mnogochlenami funktsii, udovletvoryayuschikh usloviyu Lipshitsa”, Izv. AN SSSR. Ser. matem., 10 (1946), 295–322 | MR | Zbl

[10] Dzyazyk V. K., “O konstruktivnoi kharakteristike funktsii, udovletvoryayuschikh usloviyu $\operatorname{Lip}\alpha(0\alpha1)$ na konechnom otrezke veschestvennoi osi”, Izv. AN SSSR. Ser. matem., 20 (1956), 623–642

[11] Timan A. F., “Usilenie teoremy Dzheksona o nailuchshem priblizhenii nepreryvnykh funktsii mnogochlenami na konechnom otrezke veschestvennoi osi”, Dokl. AN SSSR, 78:1 (1951), 17–20 | MR | Zbl

[12] Dolzhenko E. P., Sevastyanov E. A., “O priblizheniyakh v khausdorfovoi metrike”, Dokl. AN SSSR, 226:4 (1976), 768–770 | MR | Zbl

[13] Dolzhenko E. P., Sevastyanov E. A., “O priblizheniyakh funktsii v khausdorfovoi metrike posredstvom kusochno-monotonnykh (v chastnosti ratsionalnykh) funktsii”, Matem. sb., 101:4 (1976), 508–541 | MR | Zbl