Geodesic
Exact Constants in Jackson Inequalities for Periodic Differentiable Functions in the Space~$L_\infty$
Matematičeskie zametki, Tome 96 (2014) no. 2, pp. 277-284.

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It is proved that, in the space ${L }_\infty[0,2\pi]$, the following equalities hold for all $k=0,1,2,\dots$, $n\in\mathbb N$, $r=1,3,5,\dots$, $\mu\ge r$: $$ \sup_{\substack{f\in {L }_\infty^r\\ f\ne\operatorname{const}}} \frac{{E}_{n-1}(f)}{\omega(f^{(r)},\pi/(n(2k+1)))}= \sup_{\substack{f\in {L }_\infty^r\\ f\ne\operatorname{const}}} \frac{{E}_{n,\mu}(f)}{\omega(f^{(r)},\pi/(n(2k+1)))}= \frac{\|\psi_{r,2k+1}\|}{2n^r}\mspace{2mu}, $$ where ${E}_{n-1}(f)$ and ${E}_{n,\mu}(f)$ are the best approximations of $f$ by, respectively, trigonometric polynomials of degree $n-1$ and $2\pi$-periodic splines of minimal deficiency of order $\mu$ with $2n$ equidistant nodes, $\omega(f^{(r)},h)$ is the modulus of continuity of $f^{(r)}$, $\psi_{r,2k+1}$ is the $r$th periodic integral of the special function $\psi_{0,2k+1}$, which is odd and piecewise constant on the partition $j\pi/ (2k+1)$, $j\in\mathbb Z$. For $k=0$, this result was obtained earlier by Ligun.
Keywords: Jackson inequality, exact constant in the Jackson inequality, $2\pi$-periodic function, the space $L_\infty$, best approximation by trigonometric polynomials, best approximation by $2\pi$-periodic splines, Jackson constant
Mots-clés : Favard constant. 
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S. A. Pichugov. Exact Constants in Jackson Inequalities for Periodic Differentiable Functions in the Space~$L_\infty$. Matematičeskie zametki, Tome 96 (2014) no. 2, pp. 277-284. http://geodesic.mathdoc.fr/item/MZM_2014_96_2_a11/

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

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

[3] A. A. Ligun, “O tochnykh konstantakh priblizheniya differentsiruemykh periodicheskikh funktsii”, Matem. zametki, 14:1 (1973), 21–30 | MR | Zbl

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

[5] V. V. Zhuk, “O nekotorykh tochnykh neravenstvakh mezhdu nailuchshimi priblizheniyami i modulyami nepreryvnosti”, Sib. matem. zhurn., 12:6 (1971), 1283–1291 | MR | Zbl