Böttcher, Albrecht ; Grudsky, Sergei M. ; Maksimenko, Egor A. : Pushing the Envelope of the Test Functions in the Szegö and Avram-Parter Theorems

Author(s):

Böttcher, Albrecht
Grudsky, Sergei M.
Maksimenko, Egor A.

Title:

Pushing the Envelope of the Test Functions in the Szegö and Avram-Parter Theorems

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 24, 2007

Mathematics Subject Classification:

47B35			[ Toeplitz operators, Hankel operators, Wiener-Hopf operators ]
15A15			[ Determinants, permanents, other special matrix functions ]

Abstract:

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 that the theorems hold for arbitrary convex and even only essentially convex test functions, which includes all admissible test functions known so far.

Keywords:

Toeplitz matrix, eigenvalue, singular value, test function, asymptotic distribution

Language:

English

Publication time:

11 / 2007