On the optimal Lebesgue constants for polynomial interpolation (Q579550)

Let \(X=\{x_{kn}\}\), \(k=1,2,...,n\); \(n=1,2,...\), be any triangular matrix with \[ -1=x_{n+1,n}\leq x_{nn}<x_{n- 1,n}<...<x_{2n}<x_{1n}\leq x_{0n}=1. \] The behavior of the Lebesgue function \(\lambda_ n(X,x):=\sum^{n}_{k=1}| \ell_{kn}(X,x)|\) and of the Lebesgue constant \(\lambda_ n(X):=\max_{-1\leq x\leq 1}\lambda_ n(X,x),\) is of fundamental importance in the study of Lagrange interpolation. If \(\lambda^*_ n:=\min_{x}\lambda_ n(x)\), \textit{P. Erdős} proved [ibid. 12, 235-244 (1961; Zbl 0098.004)] that \((2/n) \lambda n n-c_ 1\leq \lambda^*_ n\leq (2/\pi) \ell n+c_ 2\) where \(c_ 1,c_ 2\) are constants. The main result of the author is: \[ \frac{\text{const}}{(\ell n\;n)^{1/3}}> \lambda^*_ n-\frac{2}{\pi}\ell n\-n-\chi\geq \begin{cases} \frac{\pi}{18n^ 2}+O(n^{-4}),& n=2m \\ -\frac{2}{\pi n}+O(n^{-2}),& n=2m+1 \end{cases} \] where \(\chi:=(2/\pi)(\gamma +\ell n(4/\pi))\). He also gives a summary of the earlier known results in this connection.