RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

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

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.

DOI: https://doi.org/10.4213/mzm10339

Full text: PDF file (477 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.51
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
\Bibitem{Run14} \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 \mathnet{http://mi.mathnet.ru/mz10339} \crossref{https://doi.org/10.4213/mzm10339} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3306227} \elib{http://elibrary.ru/item.asp?id=21826514} \transl \jour Math. Notes \yr 2014 \vol 95 \issue 6 \pages 833--842 \crossref{https://doi.org/10.1134/S0001434614050289} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000338338200028} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84903388587} 

• http://mi.mathnet.ru/eng/mz10339
• https://doi.org/10.4213/mzm10339
• http://mi.mathnet.ru/eng/mz/v95/i6/p899

 SHARE:

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. S. Yu. Artamonov, “Direct Jackson-Type Estimate for the General Modulus of Smoothness in the Nonperiodic Case”, Math. Notes, 97:5 (2015), 811–814
2. S. Yu. Artamonov, “Quality of Approximation by Fourier Means in Terms of General Moduli of Smoothness”, Math. Notes, 98:1 (2015), 3–10
3. K. V. Runovskii, “Approximation by Fourier Means and Generalized Moduli of Smoothness”, Math. Notes, 99:4 (2016), 564–575
4. S. Yu. Artamonov, “Nonperiodic Modulus of Smoothness Corresponding to the Riesz Derivative”, Math. Notes, 99:6 (2016), 928–931
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
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
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
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
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
•  Number of views: This page: 326 Full text: 96 References: 43 First page: 30