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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     author = {V. A. Kofanov},
     title = {Approximation in the mean of classes of differentiable functions by algebraic polynomials},
     journal = {Izvestiya. Mathematics },
     pages = {353--365},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {1984},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1984_23_2_a5/}
}
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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/