New answers to problem 24 of P. Turán (Q1324779)

This paper gives some new results about the problem 24 proposed by \textit{P. Turán} [J. Approximation Theory 29, 23-89 (1980; Zbl 0454.41001)]. Let \(f\in C[-1,1]\), \(\|\cdot\|\) stands for the sup-norm. Given a triangular matrix \(X\) of nodes \(\{-1\leq x_{n1}< x_{n2}<\cdots< x_{nn}\leq 1: n=1,2,3,\dots\}\). Denote by \(L_ n(f,x)\) and \(H_ n(f,x)\) the Lagrange interpolating polynomial with degree \(n-1\) for \(f\) at nodes \(\{x_{n1},\dots,x_{nn}\}\) and the Hermite interpolating polynomial with degree \(2n-1\) for \(f\) at double nodes \(\{x_{n1},\dots, x_{nn}\}\) respectively. In his problem 24, P. Turán asks: is it true that, for any matrix \(X\) satisfying \(\lim_{n\to \infty} \| H_ n(f)- f\|= 0\) for all \(f\in C[-1,1]\), we have \(\lim_{n\to\infty}\| L_ n(f)- f\|= 0\) for all \(f\in C^ 1[- 1,1]\)? P. Vertesi and the author considered this problem. They obtained some answers. In this paper the author gives some new results about the same problem, improving his former results. Among other things, the author proves that, if \(\mu_ n= O(1)\), then the question is answered affirmatively for all \(f\in \text{Lip }\alpha\) with \(\alpha> {1\over 4}(\sqrt{17}- 1)\), where \(\mu_ n\) is some quantity relating to \(X\), whose definition is too long to be cited here.