On a problem of P. Erdős concerning polynomials and interpolation
In 1980 year, P. Erdős suggested the following problem concerning polynomials and interpolation: Let \[ \begin{gathered} -1\leq x_1<x_2<\cdots<x_{n-1}<x_n\leq 1, \quad n\geq 2, \\ \omega_n(x)=\prod^n_{i=1}(x-x_i)\quad \text{be given}. \end{gathered} \] Determine \(\tau_n:=\max_{x_1,\dots,x_n}\), \(\min_{1\leq k\leq n} | \omega_n'(x_k)| \). In theorem 1 the author gives an estimate of \(\tau_n\). Theorem 1: We have \[ \frac{n-1}{2^{n-2}}\leq \tau_n\leq\frac{n-2} {2^{n-2}\left(\cos \frac{\pi}{2(n-1)}\right)^{n-1}}, \quad n\geq 3, \] and \(\lim_{n\to\infty}\frac{n-1}{2^{n-2}\tau_n}=1\). In theorem 2 the author gives necessary conditions for a solution of the above extremal problem.
scientific article; zbMATH DE number 1336325