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Mat. Zametki, 2014, Volume 95, Issue 6, Pages 899–910 (Mi mz10339)  

This article is cited in 9 scientific papers (total in 9 papers)

A Direct Theorem of Approximation Theory for a General Modulus of Smoothness

K. V. Runovskii

Lomonosov Moscow State University, Chernomorsky Branch

Abstract: We introduce the notion of general modulus of smoothness in the spaces $L_p$ of $2\pi$-periodic $p$th-power integrable functions; in these spaces, the coefficients multiplying the values of a given function at the nodes of the uniform lattice are the Fourier coefficients of some $2\pi$-periodic function called the generator of the modulus. It is shown that all known moduli of smoothness are special cases of this general construction. For the introduced modulus, in the case $1 \le p \le {+\infty}$, we prove a direct theorem of approximation theory (a Jackson-type estimate). It is shown that the known Jackson-type estimates for the classical moduli, the modulus of positive fractional order, and the modulus of smoothness related to the Riesz derivative are its direct consequences. We also obtain a universal structural description of classes of functions whose best approximations have a certain order of convergence to zero.

Keywords: Jackson-type estimate, modulus of smoothness, $2\pi$-periodic $p$th-power integrable function, Fourier mean, Hölder's inequality, Fourier coefficient.


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English version:
Mathematical Notes, 2014, 95:6, 833–842

Bibliographic databases:

UDC: 517.51
Received: 07.06.2013
Revised: 02.11.2013

Citation: K. V. Runovskii, “A Direct Theorem of Approximation Theory for a General Modulus of Smoothness”, Mat. Zametki, 95:6 (2014), 899–910; Math. Notes, 95:6 (2014), 833–842

Citation in format AMSBIB
\by K.~V.~Runovskii
\paper A Direct Theorem of Approximation Theory for a General Modulus of Smoothness
\jour Mat. Zametki
\yr 2014
\vol 95
\issue 6
\pages 899--910
\jour Math. Notes
\yr 2014
\vol 95
\issue 6
\pages 833--842

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    This publication is cited in the following articles:
    1. S. Yu. Artamonov, “Direct Jackson-Type Estimate for the General Modulus of Smoothness in the Nonperiodic Case”, Math. Notes, 97:5 (2015), 811–814  mathnet  crossref  crossref  mathscinet  isi  elib
    2. S. Yu. Artamonov, “Quality of Approximation by Fourier Means in Terms of General Moduli of Smoothness”, Math. Notes, 98:1 (2015), 3–10  mathnet  crossref  crossref  mathscinet  isi  elib
    3. K. V. Runovskii, “Approximation by Fourier Means and Generalized Moduli of Smoothness”, Math. Notes, 99:4 (2016), 564–575  mathnet  crossref  crossref  mathscinet  isi  elib
    4. S. Yu. Artamonov, “Nonperiodic Modulus of Smoothness Corresponding to the Riesz Derivative”, Math. Notes, 99:6 (2016), 928–931  mathnet  crossref  crossref  mathscinet  isi  elib
    5. K. V. Runovskii, N. V. Omel'chenko, “Mixed Generalized Modulus of Smoothness and Approximation by the “Angle” of Trigonometric Polynomials”, Math. Notes, 100:3 (2016), 448–457  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. S. B. Vakarchuk, “Best Polynomial Approximations and Widths of Classes of Functions in the Space $L_2$”, Math. Notes, 103:2 (2018), 308–312  mathnet  crossref  crossref  isi  elib
    7. S. Yu. Artamonov, “On some constructions of a non-periodic modulus of smoothness related to the Riesz derivative”, Eurasian Math. J., 9:2 (2018), 11–21  mathnet
    8. Artamonov S. Runovski K. Schmeisser H.-J., “Approximation By Bandlimited Functions, Generalized K-Functionals and Generalized Moduli of Smoothness”, Anal. Math., 45:1 (2019), 1–24  crossref  mathscinet  zmath  isi  scopus
    9. K. V. Runovskii, “Generalized Smoothness and Approximation of Periodic Functions in the Spaces $L_p$, $1<p<+\infty$”, Math. Notes, 106:3 (2019), 412–422  mathnet  crossref  crossref  isi  elib
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