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

 Mat. Zametki, 1975, Volume 18, Issue 4, Pages 515–526 (Mi mz9966)

Lebesgue's inequality in a uniform metric and on a set of full measure

K. I. Oskolkov

V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR

Abstract: Let $f$ be a continuous periodic function with Fourier sums $S_n(f)$, $E_n(f)=E_n$ be the best approximation to $f$ by trigonometric polynomials of order $n$. The following estimate is proved:
$$||f-S_n(f)||\leqslant c\sum_{\nu=n}^{2n}\frac{E_\nu}{\nu-n+1}.$$
(Here $c$ is an absolute constant.) This estimate sharpens Lebesgue's classical inequality for “fast” decreasing $E_\nu$. The sharpness of this estimate is proved for an arbitrary class of functions having a given majorant of best approximations. Also investigated is the sharpness of the corresponding estimate for the rate of convergence of a Fourier series almost everywhere.

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English version:
Mathematical Notes, 1975, 18:4, 895–902

Bibliographic databases:

UDC: 517.5

Citation: K. I. Oskolkov, “Lebesgue's inequality in a uniform metric and on a set of full measure”, Mat. Zametki, 18:4 (1975), 515–526; Math. Notes, 18:4 (1975), 895–902

Citation in format AMSBIB
\Bibitem{Osk75} \by K.~I.~Oskolkov \paper Lebesgue's inequality in a uniform metric and on a set of full measure \jour Mat. Zametki \yr 1975 \vol 18 \issue 4 \pages 515--526 \mathnet{http://mi.mathnet.ru/mz9966} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=412711} \zmath{https://zbmath.org/?q=an:0339.42001} \transl \jour Math. Notes \yr 1975 \vol 18 \issue 4 \pages 895--902 \crossref{https://doi.org/10.1007/BF01153041} 

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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
2. A. I. Syusyukalov, “On the approximation of functions in the class $C(\varepsilon)$ using means of sequences of Fourier sums”, Russian Math. (Iz. VUZ), 42:5 (1998), 76–78
3. Temlyakov V., “Nonlinear Methods of Approximation”, Found. Comput. Math., 3:1 (2003), 33–107
4. I. I. Sharapudinov, “Sobolev orthogonal polynomials generated by Jacobi and Legendre polynomials, and special series with the sticking property for their partial sums”, Sb. Math., 209:9 (2018), 1390–1417
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