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

This article is cited in 5 scientific papers (total in 5 papers)

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-а
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

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

Bibliographic databases:

UDC: 517.518.832+517.518.837
MSC: 42A10, 41A17, 42B15
Received: 11.03.2015 and 10.04.2016

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
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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  crossref  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref  crossref  adsnasa  isi
    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  mathnet  crossref  crossref  isi  elib
    5. N. V. Omel'chenko, K. V. Runovskii, “Realizations of Mixed Generalized $K$-Functionals”, Math. Notes, 107:2 (2020), 353–356  mathnet  crossref  crossref  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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