On the extended Hermite-Fejér interpolation process
Let us consider a matrix \[ (M)\quad \{x_ k^{(n)}\},\quad k=1,2,...,n,\quad n=1,2,...,\quad -1<x_ n^{(n)}<x^{(n)}_{n- 1}<...<x_ 1^{(n)}<1. \] We say that the matrix (M)\(\in K\) if (M) satisfies the following conditions: 1) (M) is \(\rho\)-normal in the Fejér sense; 2) the roots of \(\omega_ n(x)=\prod^{n}_{i=1}(x-x_ i^{(n)})\) are symmetric with respect to \(x=0\); 3) for each \(x\in (- 1,1)\) there is a sequence \(\{n_ i\}\) of integers such that \(\lim_{i\to \infty}\omega^ 2_{n_ i}(x)/\omega^ 2_{n_ i}(1)\geq C(x)>0.\) Let us define a class \(\Delta_ 1\) of continuous functions consisting of all functions \(f\in C[-1,1]\) for which there exists f'(1-0). Let \(F_ n(f,x)\) be the polynomial of degree \(2n+3\), which is uniquely determined by the conditions \(F_ n(f,x_ k^{(n)})=f(x_ k^{(n)});\) \(F_ n(f,\pm 1)=f(\pm 1),\) \(F_ n'(f,x_ k^{(n)})=F_ n'(f,\pm 1)=0,\) \(k=1,2,...,n\). It was shown by the reviewer that for \(f_ 1=x^ 2\) (resp. for \(f_ 2=| x|)\) the sequence \(\{F_ n(f_ 1,x)\}\) (resp. \(\{F_ n(f_ 2,x)\})\) does not converge to \(x^ 2\) (resp. to \(| x|)\) at any point in (- 1,1). In connection with these results the author proves the following Theorem. Let (M)\(\in K\) and \(f\in \Delta_ 1\) be even. The condition f'(1)\(\neq 0\) is necessary and sufficient for the divergence of the process \(\{F_ n(f,x)\}^{\infty}_{n=1}\) everywhere on (-1,1).
scientific article; zbMATH DE number 3983790
scientific article