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. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb., 2017, Volume 208, Number 2, Pages 70–87 (Mi msb8505)

Trigonometric polynomial approximation, $K$-functionals and generalized moduli of smoothness

K. V. Runovski

Lomonosov Moscow State University, Chernomorsky Branch, Sevastopol

Abstract: Best approximation and approximation by families of linear polynomial operators (FLPO) in the spaces $L_p$, $0<p \le +\infty$, are investigated for periodic functions of an arbitrary number of variables in terms of the generalized modulus of smoothness generated by a periodic generator which, near the origin, is assumed to be close in a certain sense to some homogeneous function of positive order. Direct and inverse theorems (Jackson- and Bernstein-type estimates) are proved; conditions on the generators are obtained under which the approximation error by an FLPO is equivalent to an appropriate modulus of smoothness. These problems are solved by going over from the modulus to an equivalent $K$-functional. The general results obtained are applied to classical objects in the theory of approximation and smoothness. In particular, they are applied to the methods of approximation generated by Fejér, Riesz and Bochner-Riesz kernels, and also to the moduli of smoothness and $K$-functionals corresponding to the conventional, Weyl and Riesz derivatives and to the Laplace operator.
Bibliography: 24 titles.

Keywords: family of linear polynomial operators, best approximation, modulus of smoothness, $K$-functional, Jackson- and Bernstein-type estimates.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-01236-à This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 15-01-01236-a).

DOI: https://doi.org/10.4213/sm8505

Full text: PDF file (598 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2017, 208:2, 237–254

Bibliographic databases:

UDC: 517.518.832+517.518.837
MSC: 42A10, 41A17, 42B15

Citation: K. V. Runovski, “Trigonometric polynomial approximation, $K$-functionals and generalized moduli of smoothness”, Mat. Sb., 208:2 (2017), 70–87; Sb. Math., 208:2 (2017), 237–254

Citation in format AMSBIB
\Bibitem{Run17} \by K.~V.~Runovski \paper Trigonometric polynomial approximation, $K$-functionals and generalized moduli of smoothness \jour Mat. Sb. \yr 2017 \vol 208 \issue 2 \pages 70--87 \mathnet{http://mi.mathnet.ru/msb8505} \crossref{https://doi.org/10.4213/sm8505} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3608038} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2017SbMat.208..237R} \elib{http://elibrary.ru/item.asp?id=28172163} \transl \jour Sb. Math. \yr 2017 \vol 208 \issue 2 \pages 237--254 \crossref{https://doi.org/10.1070/SM8505} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000401433200004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85017767464} 

• http://mi.mathnet.ru/eng/msb8505
• https://doi.org/10.4213/sm8505
• http://mi.mathnet.ru/eng/msb/v208/i2/p70

 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. N. V. Omel'chenko, “On the $K$-Functional for the Mixed Generalized Modulus of Smoothness”, Math. Notes, 103:2 (2018), 319–322
2. 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
3. A. G. Baskakov, V. E. Strukov, I. I. Strukova, “Harmonic analysis of functions in homogeneous spaces and harmonic distributions that are periodic or almost periodic at infinity”, Sb. Math., 210:10 (2019), 1380–1427
4. 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
5. N. V. Omel'chenko, K. V. Runovskii, “Realizations of Mixed Generalized $K$-Functionals”, Math. Notes, 107:2 (2020), 353–356
•  Number of views: This page: 318 Full text: 13 References: 45 First page: 33