Geodesic
Approximation in the mean of classes of differentiable functions by algebraic polynomials
Izvestiya. Mathematics , Tome 23 (1984) no. 2, pp. 353-365.

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The exact values $E_n(W^r_L)_L$ are found for the best approximations in the mean of the function classes $$W^r_L=\{f:f^{(r-1)}\text{ is absolutely continuous, }\|f^{(r)}\|_L\leqslant1\},\qquad r =2,3,\dots,$$ by algebraic polynomials of degree at most $n$ on the interval $[-1,1]$. It is proved that $E_n(W^r_L)_L$ coincides with the uniform norm of the perfect spline $$ \frac1{r!}\biggl[(x+1)^r+2\sum^{n+1}_{i=1}(-1)^i(x-x_i)^r_+\biggr] $$ with nodes $x_i=-\cos\frac{i\pi}{n+2}$. Bibliography: 6 titles. 
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V. A. Kofanov. Approximation in the mean of classes of differentiable functions by algebraic polynomials. Izvestiya. Mathematics , Tome 23 (1984) no. 2, pp. 353-365. http://geodesic.mathdoc.fr/item/IM2_1984_23_2_a5/

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[2] Kofanov V. A., “Tochnye znacheniya nailuchshikh priblizhenii algebraicheskimi mnogochlenami v srednem klassov $W_L^r$ ($r=1,2$)”, Issledovaniya po sovremennym problemam summirovaniya i priblizheniya funktsii i ikh prilozheniya, DGU, Dnepropetrovsk, 1977, 18–20

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