Pushing the envelope of the test functions in the Szegö and Avram-Parter theorems
type: Item
instance of: scholarly article
title: Pushing the envelope of the test functions in the Szegö and Avram-Parter theorems
zbMATH Open document ID: 1153.15012
author: A. Böttcher
author: S. M. Grudsky
author: E. A. Maximenko
published in: Linear Algebra and its Applications
publication date: 2008-06-09T00:00:00Z
review text: Let \(a\) be a complex-valued function in \(L^1 (0, 2\pi )\) and let \(a_n\) be the \(n^{th}\) Fourier coefficient of \(a\), and denote by \(T_n (a)\) the \(n\times n\) Toeplitz matrix \((a_{ij} ), 1\leq i, j\leq n\). All the matrices \(T_n (a)\) are Hermitian iff \(a\) is real-valued. \textit{G. Szegö} [Math. Z. 6, 167--202 (1920); 167--190 (1921) (1920, 1921; JFM 48.0376.03)] has shown that \[ \lim_{n \rightarrow \infty}\frac1n \sum^{n}_{j=1}F(\lambda_{j} (T_n (a))) =\frac{1}{2\pi} \int^{2\pi}_{0} F(a(\theta )) \,d\theta \] under certain conditions on \(a\) and \(F\) including that \(a\) be real-vaued. Here \(\lambda_{1} (A)\leq\lambda_{2} (A)\leq\ldots \leq \lambda_{n} (A)\) are the eigenvalues of a Hermitian matrix \(A\). On the other hand, Avram-Parter type theorems [\textit{F. Avram}, Probab. Theory Relat. Fields 79, No. 1, 37--45 (1988; Zbl 0648.60043); \textit{S. V. Parter}, Linear Algebra Appl. 80, 115--130 (1986; Zbl 0601.15006)] state that under some assumptions on \(a\) and \(F\) we have \[ \lim_{n \rightarrow\infty}\frac1n \sum^{n}_{j=1}F (s_j (T_n (a))) =\frac{1}{2\pi}\int^{2\pi}_{0} F(| a(\theta| ))\,d\theta \] where \(s_1 (A)\leq s_2 (A)\leq \ldots \leq s_n (A)\) are singular values of an \(n\times n\) matrix \(A\). The authors of the paper under review consider the question ``are these theorems true whenever they make sense?'' and they give a negative answer to it. They show that Szegö and Avram-Parter theorems that give the limit of the arithmetic mean of the values of certain test functions \((F )\) at the eigenvalues of Hermitian Toeplitz matrices in Szegö theorems type and the singular values of arbitrary Toeplitz matrices in Avram-Parter theorems as the matrix dimension goes to infinity are not valid for all continuous, nonegative, and monotonously increasing test functions. They also obtain general results that include all versions known so far.
reviewed by: Manouchehr Misaghian
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 5285782
zbMATH Keywords: Toeplitz matrix
zbMATH Keywords: eigenvalue
zbMATH Keywords: singular value
zbMATH Keywords: test function
zbMATH Keywords: asymptotic distribution
MaRDI profile type: Publication
full work available at URL: j.laa.2008.02.031
OpenAlex ID: W2139504216
cites work: On bilinear forms in Gaussian random variables and Toeplitz matrices
cites work: Introduction to large truncated Toeplitz matrices
cites work: Some Problems Concerning the Test Functions in the Szegö and Avram-Parter Theorems
cites work: Spectral properties of block multilevel Hankel matrices
cites work: Q5512384
cites work: Q3247378
cites work: Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix
cites work: On the distribution of the singular values of Toeplitz matrices
cites work: Q4788718
cites work: On unitarily invariant norms of matrix-valued linear positive operators.
cites work: Some Results on Matrices with Prescribed Diagonal Elements and Singular Values
cites work: Singular Values, Diagonal Elements, and Convexity
cites work: A note on the spectral distribution of toeplitz matrices
cites work: A unifying approach to some old and new theorems on distribution and clustering
cites work: Spectra of multilevel Toeplitz matrices: Advanced theory via simple matrix relationships
cites work: Distribution of eigenvalues and singular values of Toeplitz matrices under weakened conditions on the generating function
DOI: 10.1016/J.LAA.2008.02.031
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description: scientific article; zbMATH DE number 5285782