Geodesic
Sharp Constant in Jackson's Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions
Matematičeskie zametki, Tome 93 (2013) no. 6, pp. 932-938.

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It is proved that, in the space $\mathrm{C}_{2\pi}$, for all $k,n\in\mathbb N$, $n>1$, the following inequalities hold: $$ \biggl(1-\frac {1}{2n}\biggr)\frac{k^2+1}{2}\le \sup_{\substack{f\in \mathrm{C}_{2\pi}\\ f\ne\mathrm{const}}} \frac{{e}_{n-1}(f)}{\omega_2(f,\pi/(2nk))}\le \frac{k^2+1}{2}\mspace{2mu}. $$ where ${e}_{n-1}(f)$ is the value of the best approximation of $f$ by trigonometric polynomials and $\omega_2(f,h)$ is the modulus of smoothness of $f$. A similar result is also obtained for approximation by continuous polygonal lines with equidistant nodes.
Keywords: Jackson's inequality, periodic function, trigonometric polynomial, modulus of smoothness, polygonal line, Steklov mean
Mots-clés : Favard sum. 
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S. A. Pichugov. Sharp Constant in Jackson's Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions. Matematičeskie zametki, Tome 93 (2013) no. 6, pp. 932-938. http://geodesic.mathdoc.fr/item/MZM_2013_93_6_a10/

[1] N. P. Korneichuk, “Tochnaya konstanta v teoreme Dzheksona o nailuchshem ravnomernom priblizhenii nepreryvnykh periodicheskikh funktsii”, DAN SSSR, 145:3 (1962), 514–515 | MR | Zbl

[2] N. P. Korneichuk, “O tochnoi konstante v neravenstve Dzheksona dlya nepreryvnykh periodicheskikh funktsii”, Matem. zametki, 32:5 (1982), 669–674 | MR | Zbl

[3] N. I. Chernykh, “O neravenstve Dzheksona v $L_2$”, Priblizhenie funktsii v srednem, Sbornik rabot, Tr. MIAN SSSR, 88, 1967, 71–74 | MR | Zbl

[4] N. I. Chernykh, “O nailuchshem priblizhenii periodicheskikh funktsii trigonometricheskimi polinomami v $L_2$”, Matem. zametki, 2:5 (1967), 513–522 | MR | Zbl

[5] N. I. Chernykh, “Neravenstvo Dzheksona v $L_p(0,2\pi)$ $(1\le p2)$ s tochnoi konstantoi”, Trudy Vsesoyuznoi shkoly po teorii funktsii (Miass, iyul 1989 g.), Tr. MIAN, 198, Nauka, M., 1992, 232–241 | MR | Zbl

[6] V. V. Shalaev, “K voprosu o priblizhenii nepreryvnykh periodicheskikh funktsii trigonometricheskimi polinomami”, Issledovaniya po sovremennym problemam summirovaniya i priblizheniya funktsii i ikh prilozheniya, Dnepropetrovsk. un-t, Dnepropetrovsk, 1979, 39–43

[7] N. P. Korneichuk, Tochnye konstanty v teorii priblizheniya, Nauka, M., 1987 | MR | Zbl

[8] A. A. Ligun, “Tochnye konstanty v neravenstvakh tipa Dzheksona”, Spetsialnye voprosy teorii priblizhenii i optimalnogo upravleniya raspredelennymi sistemami, Vischa shkola, Kiev, 1990, 3–74

[9] V. V. Zhuk, “O nekotorykh neravenstvakh mezhdu nailuchshimi priblizheniyami i modulyami nepreryvnosti”, Vestn. Leningradsk. un-ta. Ser. 1. Matem., mekh., astron., 1974, no. 1, 21–26 | MR | Zbl

[10] V. V. Zhuk, Approksimatsiya periodicheskikh funktsii, Izd-vo Leningradsk. un-ta, Leningrad, 1982 | MR | Zbl

[11] A. G. Babenko, Yu. V. Kryakin, “Integralnoe priblizhenie kharakteristicheskoi funktsii intervala i neravenstvo Dzheksona v $C(\mathbb T)$”, Tr. IMM UrO RAN, 15, no. 1, 2009, 59–65

[12] N. P. Korneichuk, Splainy v teorii priblizheniya, Nauka, M., 1984 | MR | Zbl