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Mat. Zametki, 1975, Volume 18, Issue 1, Pages 77–90 (Mi mz7628)  

This article is cited in 1 scientific paper (total in 1 paper)

Approximation of integrable functions by linear methods almost everywhere

T. V. Radoslavova

V. A. Steklov Mathematical Institute, USSR Academy of Sciences

Abstract: It is shown that $2\pi$ periodic functions whose $(r-1)$-th derivatives have bounded variation $(r>0)$ can be approximated by de La Vallée-Poussin $\sigma_{n,m}(an\le m=m(n)\le An, 0<a<A<1)$ at almost all points with a rate $o(n^{–r})$. For functions belonging to the class $\operatorname{Lip}(\alpha,L)(0<\alpha<1)$, any natural $N$, and a positive $\varepsilon>0$, we have almost everywhere
$$ |f(x)-\sigma_{n,m}(f;x)|\le c(f,x)n^{-\alpha}\ln n…\ln_N^{1+\varepsilon}n, $$
where $\ln_kx=\underbrace{\ln…\ln x}_k(k=1,2,…)$. For any triangular method of summation $T$ with bounded coefficients we construct functions belonging to $\operatorname{Lip}(\alpha,L)(0<\alpha<1)$ and such that almost everywhere,
$$ \varlimsup_{n\to\infty}|f(x)-\tau_n(f;x)|n^\alpha(\ln n…\ln_Nn)^{-\alpha}=\infty, $$
where the $\tau_n(f;x)$ are the means of the method $T$.

Full text: PDF file (832 kB)

English version:
Mathematical Notes, 1975, 18:1, 628–636

Bibliographic databases:

UDC: 517.5
Received: 04.11.1974

Citation: T. V. Radoslavova, “Approximation of integrable functions by linear methods almost everywhere”, Mat. Zametki, 18:1 (1975), 77–90; Math. Notes, 18:1 (1975), 628–636

Citation in format AMSBIB
\Bibitem{Rad75}
\by T.~V.~Radoslavova
\paper Approximation of integrable functions by linear methods almost everywhere
\jour Mat. Zametki
\yr 1975
\vol 18
\issue 1
\pages 77--90
\mathnet{http://mi.mathnet.ru/mz7628}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=382968}
\zmath{https://zbmath.org/?q=an:0312.41011}
\transl
\jour Math. Notes
\yr 1975
\vol 18
\issue 1
\pages 628--636
\crossref{https://doi.org/10.1007/BF01461144}


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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. K. I. Oskolkov, “Approximation properties of summable functions on sets of full measure”, Math. USSR-Sb., 32:4 (1977), 489–514  mathnet  crossref  mathscinet  zmath  isi
  • Математические заметки Mathematical Notes
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