On the asymptotic estimates of the best approximations of differentiable functions by algebraic polynomials in the space \(L_ 1\) (Q1336010)

Denote by \(W^ r_ \infty\) the space of smooth functions \(\{f: [-1, 1]\to \mathbb{R}\): \({\mathbf f}^{(r- 1)}\) is absolutely continuous \(\|{\mathbf f}^{(r)}\|_ \infty\leq 1\}\), \({\mathbf r}= 1,2,\dots\) and by \(E_ n(f)_ 1\) the best approximation of \(f\in W^ r_ \infty\) by polynomials of degree at most \(n\) in the norm of \(L_ 1\). Main result: If \(r= 1,2,\dots\), and \(n> r- 1\), then \[ E_ n(W^ r_ \infty)_ 1= \sup_{f\in W^ r_ \infty} E_ n(f)_ 1= {2\over \pi} B\Biggl({1\over 2}, {r\over 2}+ 1\Biggr)\;{K_{r+ 1}\over (n+ 1)^ r}+ o\Biggl({1\over n^ r}\Biggr), \] where \(B(x, y)\) is the Euler integral of the first kind and \(K_ r\) are the Favard constants.