Extremal polynomials on the $n$-grid

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 $\mathrm{\parallel }p{\mathrm{\parallel }}_{[-1,1]}$ 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 $\frac{1}{n}\log \mathrm{\parallel }{p}_n^{\ast }{\mathrm{\parallel }}_{[-1,1]}$ as $n\to \mathrm{\infty }$. The interest in this limit comes from a connection with an impossibility theorem on stable approximation on the $n$-grid.