Bounds for polynomials with a unit discrete norm

Abstract

Let $E$ be the set of $N$ equidistant points in $(-1,1)$ and $\mathbb{P}_n(E)$ be the set of all polynomials $P$ of degree $\le n$ with $\max\{|P(\zeta)|,\zeta\in E\}\le 1$. We prove that \[ K_{n,N}(x)=\max_{P\in\mathbb{P}_n(E)}|P(x)|\le C\log\frac\pi{\arctan(\frac Nn\sqrt{r^2-x^2})}, \] \[ |x|\le r:=\sqrt{1-n^2/N^2} \] where $n\lt N$ and $C$ is an absolute constant. The result is essentially sharp. Bounds for $K_{n,N}(z)$, $z\in\mathbb{C}$, uniform for $n\lt N$, are also obtained.

The method of proof of those results is a general one. It allows one to obtain sharp, or sharp up to a $\log N$ factor, bounds for $K_{n,N}$ under rather general assumptions on $E$ ($\#E=N$). A “model” result is announced for a class of sets $E$. Main components of the method are discussed in some detail in the process of investigating the case of equally spaced points.