A remark to the extended Hermite-Fejér type interpolation of higher order (Q1085354)

For \(f\in C[-1,1]\) and any natural n we consider the algebraic polynomials \(\delta _{n,r}(f,x)\), \(r=0,1,2,3\). \(\delta _{u,r}(f,x)\) is the polynomial of degree \(4n+2r+1\) which is uniquely determined by the conditions: \(\delta _{n,r}(f,x_{kn})=f(x_{kn}),\) \(S^{(j)}_{n,r}(f,x_{kn})=0,\) \(j=1,2,3\), \(k=1,2,...,n\). \(\delta _{n,r}(f,\pm 1)=f(\pm 1)\), \(S^{(i)}_{n,r}(f,\pm 1)=0\), \(i=1,...,r\), \(x_{kn}=\cos (2k-1)\pi /2n.\) The author states the theorem: the interpolation process \(\{\delta _{n,r}(f,x)\}^{\infty}_{n=1}\), \((r=1,2,3)\), \(n=1,2,...\), constructed with the nodes \(\{\pm 1\}\cup \{x_{kn}\}^ n_{k=1}\), \(n=1,2,..\). for a function f converges to f uniformly in [-1,1] provided \(| f^{(r+1)}(x)| \leq C_ 1,\) \(| x| \leq 1\), and \(f'(\pm 1)=f''(\pm 1)=f^{(r)}(\pm 1)=0\) \((C_ 1\) is an absolute constant).