[Les théorèmes de Szegő et d'Avram–Parter pour des fonctions test générales]
Les théorèmes de Szegő et d'Avram–Parter donnent la limite de la moyenne arithmetique des valeurs d'une ‘bonne’ fonction test prise en les valeurs propres de matrices de Toeplitz hermitiennes et en les valeurs singulières de matrices de Toeplitz arbitraires quand la dimension de la matrice tend vers l'infini. Nous montrons que, de manière surprenante, ces théorèmes ne sont pas valables pour une fonction test continue, positive et croissante arbitraire, alors même que leur énoncé a bien un sens. En revanche, nous prouvons les deux théorémes sous une forme générale qui inclut toutes les versions connues jusqu'ici.
The Szegő and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. We show that, surprisingly, these theorems are not true for every continuous, nonnegative, and monotonously increasing test function and thus do not hold whenever they make sense. On the other hand, we prove the two theorems in a general form which includes all versions known so far.
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@article{CRMATH_2008__346_13-14_749_0,
author = {B\"ottcher, Albrecht and Grudsky, Sergei M. and Maksimenko, Egor A.},
title = {The {Szeg\H{o}} and {Avram{\textendash}Parter} theorems for general test functions},
journal = {Comptes Rendus. Math\'ematique},
pages = {749--752},
publisher = {Elsevier},
volume = {346},
number = {13-14},
year = {2008},
doi = {10.1016/j.crma.2008.06.002},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.06.002/}
}
TY - JOUR AU - Böttcher, Albrecht AU - Grudsky, Sergei M. AU - Maksimenko, Egor A. TI - The Szegő and Avram–Parter theorems for general test functions JO - Comptes Rendus. Mathématique PY - 2008 SP - 749 EP - 752 VL - 346 IS - 13-14 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.06.002/ DO - 10.1016/j.crma.2008.06.002 LA - en ID - CRMATH_2008__346_13-14_749_0 ER -
%0 Journal Article %A Böttcher, Albrecht %A Grudsky, Sergei M. %A Maksimenko, Egor A. %T The Szegő and Avram–Parter theorems for general test functions %J Comptes Rendus. Mathématique %D 2008 %P 749-752 %V 346 %N 13-14 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.06.002/ %R 10.1016/j.crma.2008.06.002 %G en %F CRMATH_2008__346_13-14_749_0
Böttcher, Albrecht; Grudsky, Sergei M.; Maksimenko, Egor A. The Szegő and Avram–Parter theorems for general test functions. Comptes Rendus. Mathématique, Tome 346 (2008) no. 13-14, pp. 749-752. doi : 10.1016/j.crma.2008.06.002. http://www.numdam.org/articles/10.1016/j.crma.2008.06.002/
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