[Distributions asymptotiques de zéros et orthogonalité]
On développe une nouvelle méthode pour établir la distribution asymptotique des zéros de divers polynômes orthogonaux sur un segment. Cette méthode exploite de manière directe les relations d'orthogonalité. Nous l'illustrons dans quatre cas : l'orthogonalité classique par rapport à une mesure positive, l'orthogonalité non-Hermitienne par rapport à une mesure complexe, et l'orthogonalité non-linéaire intervenant en approximation rationnelle, tout d'abord dans le cas d'une mesure positive, puis dans le cas non- Hermitien.
We develop a new method to prove asymptotic zero distribution for different kinds of orthogonal polynomials. The method directly uses the orthogonality relations. We illustrate the procedure in four cases: classical orthogonality, non-Hermitian orthogonality, orthogonality in rational approximation of Markov functions and its non- Hermitian variant.
Keywords: orthogonal polynomials, zero distribution, logarithmic potential, rational approximation
Mot clés : polynômes orthogonaux, distribution des zèros, potentiel logarithmique, approximation rationnelle
@article{AIF_2005__55_5_1455_0,
author = {Baratchart, Laurent and K\"ustner, Reinhold and Totik, Vilmos},
title = {Zero distributions via orthogonality},
journal = {Annales de l'Institut Fourier},
pages = {1455--1499},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {55},
number = {5},
year = {2005},
doi = {10.5802/aif.2130},
mrnumber = {2172271},
zbl = {1076.30010},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2130/}
}
TY - JOUR AU - Baratchart, Laurent AU - Küstner, Reinhold AU - Totik, Vilmos TI - Zero distributions via orthogonality JO - Annales de l'Institut Fourier PY - 2005 SP - 1455 EP - 1499 VL - 55 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2130/ DO - 10.5802/aif.2130 LA - en ID - AIF_2005__55_5_1455_0 ER -
%0 Journal Article %A Baratchart, Laurent %A Küstner, Reinhold %A Totik, Vilmos %T Zero distributions via orthogonality %J Annales de l'Institut Fourier %D 2005 %P 1455-1499 %V 55 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2130/ %R 10.5802/aif.2130 %G en %F AIF_2005__55_5_1455_0
Baratchart, Laurent; Küstner, Reinhold; Totik, Vilmos. Zero distributions via orthogonality. Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1455-1499. doi : 10.5802/aif.2130. http://www.numdam.org/articles/10.5802/aif.2130/
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