An everywhere divergent Hermite-Fejér type interpolation process of higher order (Q1118111)

Let f(x) be a given function on [-1,1] and \(T_ n(x)\) be an nth Chebyshev polynomial of first kind. Let's consider the matrix \(T=\{x_{k,n+2}\}^ n_{k=1}\), \(n=1,2,..\). of nodes \(x_{0,n+2}=1\), \(x_{k,n+2}=\cos (k-),\) \(k=1,...,n\), \(x_{n+1,n+2}=-1\) which are roots of \((1-x^ 2)T_ n(x)\). Let \(K_ n(f,T,x)\) be polynomials of degree \(\leq 4n+7\) uniquely determined by the conditions \[ K_ n(f,T,x_{k,n+2})=f(x_{k,n+2}),\quad Kn^{(j)}(f,T,x_{k,n+2})=0,\quad k=0,1,...,n+1,\quad j=1,2,3. \] \textit{W. L. Cook} and \textit{T. H. Mills} [Bull. Aust. Math. Soc. 12, 457-465 (1975; Zbl 0294.41001)] have shown that the Hermite-Fejér type (HFT) interpolation process \(\{K_ n(f,T,x)\}\) is a divergent process at \(x=0\). The question of divergence of this process at \(x\neq 0\) was remained open. The present paper gives the complete solution of this problem: Theorem. The HFT interpolation process \(\{K_ n(f,T,x)\) constructed for \(f(x)=(1-x^ 2)^ 3\) diverges at each point in (-1,1). Moreover, for each \(x\in (-1,1)\), there exists a sequence \(\{n_ k\}^{\infty}_{k=1}\uparrow \infty\) such that \[ \lim_{k\to \infty}K_{n_ k}((1-x^ 2)^ 3,T,x)=-\frac{95}{3}(1-x^ 2)^ 3. \] The method of proof consists of constructing the polynomial \(K_ n(f,T,x)\) in terms of the polynomial \(F_ n(f,T,x)\), with use of zeros of \(T_ n(x)\).