Extremal polynomials on the $n$-grid

Abstract

The $n$-grid $E_n$ consists of $n$ equally spaced points in $[-1,1]$ including the endpoints $\pm 1$. The extremal polynomial $p_n^{\ast }$ is the polynomial that maximizes the uniform norm $\Vert p{\Vert }_{[-1,1]}$ among polynomials $p$ of degree $\le \alpha n$ that are bounded by one on $E_n$. For every $\alpha \in (0,1)$, we determine the limit of $\frac{1}{n}\log \Vert p_n^{\ast }{\Vert }_{[-1,1]}$ as $n\to \infty$. The interest in this limit comes from a connection with an impossibility theorem on stable approximation on the $n$-grid.

19 pages, no figures; correction in the proof of Lemma 2.3