Bounds for polynomials with a unit discrete norm
Annals of mathematics, Tome 165 (2007) no. 1, pp. 55-88
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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.
@article{10_4007_annals_2007_165_55,
author = {Evguenii A. Rakhmanov},
title = {Bounds for polynomials with a unit discrete norm},
journal = {Annals of mathematics},
pages = {55--88},
publisher = {mathdoc},
volume = {165},
number = {1},
year = {2007},
doi = {10.4007/annals.2007.165.55},
mrnumber = {2276767},
zbl = {1124.41014},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.165.55/}
}
TY - JOUR AU - Evguenii A. Rakhmanov TI - Bounds for polynomials with a unit discrete norm JO - Annals of mathematics PY - 2007 SP - 55 EP - 88 VL - 165 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.165.55/ DO - 10.4007/annals.2007.165.55 LA - en ID - 10_4007_annals_2007_165_55 ER -
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
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