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

This article is cited in 7 scientific papers (total in 7 papers)

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.

Full text: PDF file (553 kB)

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  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  mathscinet  isi  elib  elib
    6. Gladkaya A.V., Vinogradov O.L., “Sharp Jackson type inequalities for spline approximation on the axis”, Anal. Math., 43:1 (2017), 27–47  crossref  mathscinet  isi  scopus
    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  mathnet
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