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 Mat. Zametki, 1973, Volume 14, Issue 1, Pages 21–30 (Mi mz7199)

Exact constants of approximation for differentiable periodic functions

A. A. Ligun

Dnepropetrovsk State University

Abstract: For all odd $r$ we construct a linear operator $B_{n,r}(f)$ which maps the set of $2\pi$-periodic functions $f(t)\in X^{(r)}$ ($X^{(r)}=X^{(r)}$ or $L_1^{(r)}$) into a set of trigonometric polynomials of order not higher than $n-1$ such that
$$\sup_{f\in X^{(r)}}\frac{n^rE_n(f)_X}{\omega(f^{(r)},\pi/n)_X}=\sup_{f\in X^{(r)}}\frac{n^r\|f-B_{n,r}(f)\|_X}{\omega(f^{(r)},\pi/n)_X}=\frac{K_r}2,$$
where $X$ is the $C$ or $L_1$ metric, $E_n(f)_X$ and $\omega(f,\delta)_X$ are the best approximation by means of trigonometric polynomials of order not higher than $n-1$ and the modulus of continuity of the function $f$ in the $X$ metric, respectively; $K_r$ are the known Favard constants.

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English version:
Mathematical Notes, 1973, 14:1, 563–569

Bibliographic databases:

UDC: 517.5

Citation: A. A. Ligun, “Exact constants of approximation for differentiable periodic functions”, Mat. Zametki, 14:1 (1973), 21–30; Math. Notes, 14:1 (1973), 563–569

Citation in format AMSBIB
\Bibitem{Lig73} \by A.~A.~Ligun \paper Exact constants of approximation for differentiable periodic functions \jour Mat. Zametki \yr 1973 \vol 14 \issue 1 \pages 21--30 \mathnet{http://mi.mathnet.ru/mz7199} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=330880} \zmath{https://zbmath.org/?q=an:0281.42001} \transl \jour Math. Notes \yr 1973 \vol 14 \issue 1 \pages 563--569 \crossref{https://doi.org/10.1007/BF01095770} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. P. Korneichuk, “On extremal problems in the theory of best approximation”, Russian Math. Surveys, 29:3 (1974), 7–43
2. O. L. Vinogradov, “Sharp Jackson type inequalities for approximation of classes of convolutions by entire functions of finite degree”, St. Petersburg Math. J., 17:4 (2006), 593–633
3. O. L. Vinogradov, V. V. Zhuk, “Estimates of functionals by the second moduli of continuity of even derivatives”, J. Math. Sci. (N. Y.), 202:4 (2014), 526–540
4. S. A. Pichugov, “Exact Constants in Jackson Inequalities for Periodic Differentiable Functions in the Space $L_\infty$”, Math. Notes, 96:2 (2014), 261–267
5. O. L. Vinogradov, “Approximation estimates for convolution classes in terms of the second modulus of continuity”, Siberian Math. J., 55:3 (2014), 402–414
6. Gladkaya A.V., Vinogradov O.L., “Sharp Jackson type inequalities for spline approximation on the axis”, Anal. Math., 43:1 (2017), 27–47
7. O. L. Vinogradov, A. V. Gladkaya, “Tochnye otsenki lineinykh priblizhenii neperiodicheskimi splainami cherez lineinye kombinatsii modulei nepreryvnosti”, Issledovaniya po lineinym operatoram i teorii funktsii. 45, Zap. nauchn. sem. POMI, 456, POMI, SPb., 2017, 55–76
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