Geodesic
Bounds for polynomials with a unit discrete norm
Evguenii A. Rakhmanov1
1 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, United States
Annals of mathematics, Tome 165 (2007) no. 1, pp. 55-88.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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.
DOI : 10.4007/annals.2007.165.55

Evguenii A. Rakhmanov 1

1 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, United States 
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Evguenii A. Rakhmanov. Bounds for polynomials with a unit discrete norm. Annals of mathematics, Tome 165 (2007) no. 1, pp. 55-88. doi : 10.4007/annals.2007.165.55. http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.165.55/

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