Mathematics of the USSR-Sbornik
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Forthcoming papers
Archive
Impact factor
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






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mathematics of the USSR-Sbornik, 1977, Volume 31, Issue 2, Pages 171–189
DOI: https://doi.org/10.1070/SM1977v031n02ABEH002297
(Mi sm2648)
 

This article is cited in 14 scientific papers (total in 14 papers)

Imbedding theorems and inequalities in various metrics for best approximations

V. I. Kolyada
References:
Abstract: Let $1\leqslant p<\infty$, and let $\lambda=\{\lambda_n\}$ be a sequence of positive numbers with $\lambda_n\downarrow0$. Denote by $E_p(\lambda)$ the class of all functions $f\in L^p(0,2\pi)$ for which the best approximation by trigonometric polynomials satisfies the condition $E_n^{(p)}(f)=O(\lambda_n)$.
In this paper the relation between best approximations in different metrics is studied. Necessary and sufficient conditions are found for the imbedding $E_p(\lambda)\subset E_q(\mu)$ ($1<p<q<\infty$), where $\{\lambda_n\}$ and $\{\mu_n\}$ are positive sequences with $\lambda_n\downarrow0$ and $\mu_n\downarrow0$.
Furthermore, it is proved that the condition of P. L. Ul'yanov
$$ \sum_{n=1}^\infty n^{q/p-2}\lambda_n^q<\infty\qquad(1\leqslant p<q<\infty) $$
is not only sufficient but is also necessary for the imbedding $E_p(\lambda)\subset L^q(0,2\pi)$.
The question of imbedding $E_p(\lambda)$ in the space of continuous functions is also considered.
Bibliography: 7 titles.
Received: 31.12.1975
Bibliographic databases:
UDC: 517.5
MSC: Primary 42A08, 41A50, 46E35; Secondary 26A86
Language: English
Original paper language: Russian
Citation: V. I. Kolyada, “Imbedding theorems and inequalities in various metrics for best approximations”, Math. USSR-Sb., 31:2 (1977), 171–189
Citation in format AMSBIB
\Bibitem{Kol77}
\by V.~I.~Kolyada
\paper Imbedding theorems and inequalities in various metrics for best approximations
\jour Math. USSR-Sb.
\yr 1977
\vol 31
\issue 2
\pages 171--189
\mathnet{http://mi.mathnet.ru//eng/sm2648}
\crossref{https://doi.org/10.1070/SM1977v031n02ABEH002297}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=454492}
\zmath{https://zbmath.org/?q=an:0346.41024|0388.41015}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1977FY72200004}
Linking options:
  • https://www.mathnet.ru/eng/sm2648
  • https://doi.org/10.1070/SM1977v031n02ABEH002297
  • https://www.mathnet.ru/eng/sm/v144/i2/p195
  • This publication is cited in the following 14 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
    Statistics & downloads:
    Abstract page:509
    Russian version PDF:205
    English version PDF:14
    References:65
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024