Approximation in the mean of classes of differentiable functions by algebraic polynomials

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 $x_i=-\cos\frac{i\pi}{n+2}$.
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