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This article is cited in 8 scientific papers (total in 8 papers)
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.
Received: 13.06.1975
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
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https://www.mathnet.ru/eng/mzm9966 https://www.mathnet.ru/eng/mzm/v18/i4/p515
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Abstract page: | 329 | Full-text PDF : | 129 | First page: | 1 |
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