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.

Authors

Evguenii A. Rakhmanov

Department of Mathematics and Statistics
University of South Florida
Tampa, FL 33620
United States