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

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

Approximation of integrable functions by linear methods almost everywhere

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 $T$.

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

Bibliographic databases:

UDC: 517.5

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
\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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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
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