Geodesic
Estimates for the error of approximation of functions in $L_p^1$ by polynomials
Izvestiya. Mathematics , Tome 79 (2015) no. 2, pp. 257-287

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We find exact estimates for the error of approximation of functions in the classes $L_p^1$ by polynomials in the Haar system and partial sums of the Faber–Schauder series in the metrics of the spaces $L_p$. The error in approximating a function $f\in L_p^1$ is estimated in terms of the norms of the first derivatives $\|f^{(1)}\|_{L_p}$ and $\|f^{(1)}-\overline S^{(1)}_n(f)\|_{L_p}$. The resulting bounds are unimprovable for some values of $n$.
Keywords: Haar system of functions, Faber–Schauder system of functions, best approximation of functions by polynomials, one-sided approximation of functions by polynomials. 
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     author = {S. B. Vakarchuk and A. N. Shchitov},
     title = {Estimates for the error of approximation of functions in $L_p^1$ by polynomials},
     journal = {Izvestiya. Mathematics },
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     url = {http://geodesic.mathdoc.fr/item/IM2_2015_79_2_a2/}
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S. B. Vakarchuk; A. N. Shchitov. Estimates for the error of approximation of functions in $L_p^1$ by polynomials. Izvestiya. Mathematics , Tome 79 (2015) no. 2, pp. 257-287. http://geodesic.mathdoc.fr/item/IM2_2015_79_2_a2/