Pushing the envelope of the test functions in the Szegö and Avram–Parter theorems

The Szegö 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. The question on whether...

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Bibliographic Details
Published in:Linear algebra and its applications 2008-07, Vol.429 (1), p.346-366
Main Authors: Böttcher, Albrecht, Grudsky, Sergei M., Maksimenko, Egor A.
Format: Article
Language:English
Subjects:
Asymptotic distribution
Eigenvalue
Singular value
Test function
Toeplitz matrix
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Summary:The Szegö 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. The question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous, nonnegative, and monotonously increasing test functions. We show that, surprisingly, the answer to this question is negative. On the other hand, we prove the two theorems in a general form which includes all versions known so far.
ISSN:0024-3795
DOI:10.1016/j.laa.2008.02.031