Showing papers by "Albrecht Böttcher published in 2008"
TL;DR: In this paper, the Frobenius norm of the commutator of two matrices has been proved to be unitarily invariant, and a completely different proof of this inequality is given.
93 citations
TL;DR: It is shown that the number of orthogonal and symmetric Toeplitz matrices of a given order is finite and all these matrices are determined and a description of the set of all symmetric toeplitzer matrices whose spectrum is a prescribed doubleton is obtained.
Abstract: We show that the number of orthogonal and symmetric Toeplitz matrices of a given order is finite and determine all these matrices. In this way we also obtain a description of the set of all symmetric Toeplitz matrices whose spectrum is a prescribed doubleton.
11 citations
TL;DR: The question on whether these theorems are true whenever they make sense is essentially the same as that on whether they are valid for all continuous, nonnegative, and monotonously increasing test functions.
11 citations
TL;DR: The Szegý o and Avram-Parter theorems are not true for every continuous, nonnegative, and monotonously increasing test function and thus do not hold whenever they make sense.
9 citations
TL;DR: In this article, it was shown that if the generating function of the sequence belongs to a smoothness scale of the Holder type and if α is the smoothness parameter, then the sequence may be unbounded for α 1.
Abstract: Uniform boundedness of sequences of variable-coefficient Toeplitz matrices is a delicate problem. Recently we showed that if the generating function of the sequence belongs to a smoothness scale of the Holder type and if α is the smoothness parameter, then the sequence may be unbounded for α 1. In this paper we prove boundedness for all α > 1/2.
7 citations
TL;DR: In this paper, the problem of finding the eigenvectors associated with the extreme eigenvalues of Toeplitz matrices generated by Fisher-Hartwig symbols was simplified by using pseudomodes.
Abstract: Questions in probability and statistical physics lead to the problem of finding the eigenvectors associated with the extreme eigenvalues of Toeplitz matrices generated by Fisher-Hartwig symbols. We here simplify the problem and consider pseudomodes instead of eigenvectors. This replacement allows us to treat fairly general symbols, which are far beyond Fisher-Hartwig symbols. Our main result delivers a variety of concrete unit vectors xn such that if Tn(a) is the n × n truncation of the infinite Toeplitz matrix generated by a function a ∈ L1 satisfying mild additional conditions and λ is in the range of this function, then ‖Tn(a)xn − λxn‖ → 0 . Mathematics subject classification (2000): 47B35, 15A18, 41A80, 46N30.
5 citations
18 Dec 2008
TL;DR: In this article, the authors consider the limit of the arithmetic mean of the test functions at the eigenvalues and singular values of Toeplitz matrices as the matrix dimension increases to infinity.
Abstract: The Szego and Avram-Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues and singular values of Toeplitz matrices as the matrix dimension increases to infinity. This paper is concerned with some questions that arise when the test functions do not satisfy the known growth restrictions at infinity or when the test function has a logarithmic singularity within the range of the symbol. Several open problems are listed and accompanied by a few new results that illustrate the delicacy of the matter.
3 citations
TL;DR: In this paper, a rigorous derivation of bounds for the expected value and the variance of the spectral norm of the error in large covariance matrices is presented. But this derivation is based on a deep result by Yin et al. (Probabab. Theor. Relat. Fields 1988; 78:509−521), which gives the asymptotics of the maximal eigenvalue of a random matrix as the matrix dimension goes to infinity.
Abstract: This paper is motivated by recent studies of Huang et al. on distributed PCA and network anomaly detection and contains a rigorous derivation of bounds for the expected value and the variance of the spectral norm of the error in large covariance matrices. This derivation is based on a deep result by Yin et al. (Probab. Theor. Relat. Fields 1988; 78:509–521), which gives the asymptotics of the maximal eigenvalue of a random matrix as the matrix dimension goes to infinity. Copyright © 2007 John Wiley & Sons, Ltd.
1 citations