A few days ago, I suddenly came up with a question about polynomials: Let $x_k=-1+\dfrac{2k}{n}$, where $k=0,1,\cdots,n$, be $n+1$ equidistant nodes on $[-1,1]$. Consider polynomial $p_n$ with order less or equal than $n$ satisfying $$ |p(x_k)|\leq M, $$ where $M>0$ is a given real number. I want to know something about $$ \max_{p_n\in P_n}\max_{x\in[-1,1]}|p_n(x)|. $$ For $n=1$ it is clear that the maximum value is $M$ and for $n=2$, I believe it is $\dfrac54M$.
Moreover, I'm considering about $$ \min_{\{x_k\}\subset [-1,1]}\max_{p_n\in P_n}\max_{x\in[-1,1]}|p_n(x)|. $$ That is, for which $x_k$ can the above maximum value be minimized?