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

翻译：$n$-网格$E_n$由$[-1,1]$区间内包括端点$\pm 1$在内的$n$个等距点组成。极值多项式$p_n^*$是指在$E_n$上界于1且次数不超过$\alpha n$的多项式$p$中，最大化一致范数$\| p \|_{[-1,1]}$的多项式。对于每个$\alpha \in (0,1)$，我们确定了当$n \to \infty$时，$\frac{1}{n} \log \| p_n^*\|_{[-1,1]}$的极限。该极限的重要性源于它与$n$-网格上稳定逼近的一个不可能性定理的联系。