Comptes Rendus
Functional Analysis
The Szegő and Avram–Parter theorems for general test functions
Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 749-752.

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.

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.

Published online:
DOI: 10.1016/j.crma.2008.06.002

Albrecht Böttcher 1; Sergei M. Grudsky 2; Egor A. Maksimenko 3

1 Fakultät für Mathematik, TU Chemnitz, 09107 Chemnitz, Germany
2 Departamento de Matemáticas, CINVESTAV del I.P.N., Apartado Postal 14-740, 07000 México, D.F., Mexico
3 Department of Mathematics, Mechanics and Computer Science, Southern Federal University, 344006 Rostov-on-Don, Russia
     author = {Albrecht B\"ottcher and Sergei M. Grudsky and Egor A. Maksimenko},
     title = {The {Szeg\H{o}} and {Avram{\textendash}Parter} theorems for general test functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {749--752},
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     year = {2008},
     doi = {10.1016/j.crma.2008.06.002},
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Albrecht Böttcher; Sergei M. Grudsky; Egor A. Maksimenko. The Szegő and Avram–Parter theorems for general test functions. Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 749-752. doi : 10.1016/j.crma.2008.06.002.

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