On divergence of Hermite-Fejér interpolation to \(f(z)=z\) in the complex plane (Q1963851)

The aim of this paper is the following result: Let \(X= \{x^n_k\}\), \(n\in\{1,2,\dots\}\), \(k\in 1,\dots, n\), \(-1\leq x^{(n)}_1<\cdots< x^{(n)}_n\leq 1\), be an interpolatory matrix of nodes coinciding with the roots of the polynomials which are orthogonal on \([-1,1]\) with respect to an arbitrary summable, almost everywhere positive, weight function. Then \(\lim_{n\to\infty}|r_{2n-1}(z)|= \infty\), \(z\not\in [-1,1]\), where \(r_{2n-1}(z)= z- H_{2n-1}(z)\) and \(H_{2n-1}(f, x;z)\) are the Hermite-Fejér interpolating polynomials.