Geodesic
Lebesgue's inequality in a uniform metric and on a set of full measure
Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 515-526.

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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. 
@article{MZM_1975_18_4_a4,
     author = {K. I. Oskolkov},
     title = {Lebesgue's inequality in a uniform metric and on a set of full measure},
     journal = {Matemati\v{c}eskie zametki},
     pages = {515--526},
     publisher = {mathdoc},
     volume = {18},
     number = {4},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_18_4_a4/}
}
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K. I. Oskolkov. Lebesgue's inequality in a uniform metric and on a set of full measure. Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 515-526. http://geodesic.mathdoc.fr/item/MZM_1975_18_4_a4/