On asymptotic behavior of the best approximations of classes of differentiable functions by algebraic polynomials in mean (Q1374930)

Let \(W_p^r,r=1,2,\dots, 1\leq p\leq\infty\) be the classes of real functions defined in \([-1, 1]\) whose \((r-1)\)th derivative is absolutely continuous and \(|f^{(r)}|_p \leq 1\). As usual \(P_n\) is the set of polynomials whose degrees do not exceed \(n\). The best approximation of the classes \(W^r_p\) by algebraic polynomials \(P_n\) in the \(L_q\) norm is defined by: \[ E_n(W_p^r)_q= \sup_{f\in W_p^r}\inf_{u\in P_n}|f- u|_q, \quad 1\leq p,q\leq\infty. \] The authors announce asymptotic equalities concerning the best approximation of the function classes \(W_p^r\). For instance \[ E_n(W_p^r)_1= \left(\frac1{2\pi}\int_{-1}^1(1-t^2)^{rq/2}dt\right)^{1/q} |\varphi_{n,r}|_q+ o(1/n^r),\quad {\textstyle \frac1p+\frac 1 q}=1 \] where \(\varphi_{n,r}\) is the \(r\)-periodic integral of the function \(\text{sign} \sin(n+1) t\), whose mean value on the period is equal to zero. No proofs are given.