On the degree of approximation of the Hermite and Hermite-Fejer interpolation (Q1186327)

Let \(-1=x_{n+1}<x_ n<\dots<x_ 1<x_ 0=1\) be the \(n+2\) distinct zeros of \((1-x^ 2)P_ n(x)\) where \(P_ n\) is the Legendre polynomial of degree \(n\). With \(f\in c[-1,1]\) we associate the unique polynomial \(R_ n(f,x)\) of degree \(\leq 2n+3\) satisfying the conditions \(R_ n(f,x_ k)=f(x_ k)\), \(R_ n'(f,x_ k)=0\), \(k=0,1,\dots,n+1\). Then it is proved in the paper that \[ | R_ n(f,x)-f(x)|\leq{C \over n} \sum_{k=1}^ n\omega_ f \left({\sqrt{1-x^ 2} \over k}\right)+C \sqrt {1-x^ 2}\left({1\over n}+\sum_{k=1}^ n \omega_ f\left(1 \over k^ 2\right)\right). \] In particular if \(f\in\text{Lip} \sigma\), \({1\over 2}<\sigma<1\), then \(| R_ n(f,x)-f(x)|\leq C(1-x^ 2)^{\sigma/2}n^{1-2\sigma}\). Thus for \(f\in\text{lip} \sigma\), \({1\over2}<\sigma<1\), \(R_ n(f,\cdot)\) uniformly converges to \(f\). On the other hand for \(f(x):=(1-x^ 2)^ \sigma\), \(0<\sigma\leq{1\over 2}\), the sequence \(\{R_ n(f,x)\}\) diverges at \(x=0\). If \(f\in C^ 1[- 1,1]\) and \(F_ n(f,x)\) is the unique polynomial of degree \(\leq 2n+1\) satisfying \(F_ n(f,x_ k)=f(x_ k)\), \(k=0,\dots,n+1\), and \(F_ n'(f,x_ k)=f'(x_ k)\), \(k=1,\dots,n\), then the situation is different and we have \(| F_ n(f,x)-f(x)|\leq C{\log n \over n}E_{2n}(f')\) where \(E_{2n}(f')\) denotes the degree of best polynomial approximation to \(f'\).