On the extended Krylov-Shtaerman interpolation process (Q1117411)

Let \(x_ k^{(n)}=\cos (2k-1/2n)\pi\), \(k=1,...,n\), \(n=1,2,..\). and C be the set of all continuous functions in [-1,1]. Denote by \(H_ n(f,x)\) a polynomial of degree 2n-1, single-valued defined by the conditions \(H_ n(f,x_ k^{(n)})=f(x_ k^{(n)})\), \(H_ n'(f,x_ k^{(n)})=0\), \(k=1,2,...,n\) where \(f\in C\). As known, the process \(\{H_ n(f,x)\}\) is called the interpolation process of Hermite-Fejer. The polynomial \(P_ n(f,x)\) of degree 4n-1, single-valued defined by 4n conditions \(P_ n(f,x_ k^{(n)})=f(x_ k^{(n)})\), \(P_ n^{(i)}(f,x_ k^{(n)})=0\), \(i=1,2,3\), \(k=1,2,...,n\) where \(P_ n^{(i)}(f,x)\) is the ith derivative of \(P_ n(f,x)\), is called Krylov-Shtaerman interpolation process. This paper proves the theorem: The extended interpolational process \(\{\pi_ n(f,x)\}\) of Krylov-Shtaerman of degree \(8m+3\) which is single-valued defined as \(\pi_ n(f,x_ k^{(2m)})=f(x_ k^{(2m)}),\) \(\pi_ n^{(i)}(f,x_ k^{(2m)})=0\) \(i=1,2,3\), \(k=1,2,...,2m\), \(\pi_ n(f,0)=f(0)\), \(\pi_ n^{(i)}(f,0)=0\), \(i=1,2,3\), \(x_ k^{2m}=\cos (2k-1/4m)\pi\) converges uniformly in [-1,1] for \(f(x)=x^{2m}\), \(m\geq 2\).