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Convex Analysisの二,三の進展について

The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

/pdf/toeplitz-and-circulant-matrices-3e6lj1a9jc.pdf

Toeplitz and circulant matrices

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.

Colloquium: Area laws for the entanglement entropy

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

When working with structured matrices, structured condition numbers are more meaningful than usual condition numbers. The usual condition numbers of Toeplitz matrices may grow exponentially with the dimension of the matrix. The purpose of this note is to show that in this case the structured condition numbers do with high probability also increase exponentially, which supports the claim that one is not likely to win much on the average Toeplitz input by passing from the usual condition numbers of Toeplitz matrices to their structured counterparts. Copyright © 2004 John Wiley & Sons, Ltd.

Structured condition numbers of large Toeplitz matrices are rarely better than usual condition numbers

https://pages.pomona.edu/~sg064747/PAPERS/TDWPC.pdf

Toeplitz determinants with perturbations in the corners

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.

Orthogonal Symmetric Toeplitz Matrices

Some questions for Toeplitz operators on H p have already been answered in the preceding chapters. In particular, we settled the Fredholm theory for the operators in algℒ (H N p ) T(C N × N +H N × N ∞ ). Also notice the localization result stated in Theorem 2.96. However, many questions still remain open and it is only a small number of them which will be answered in this chapter.

Toeplitz operators on Hp

https://www.math.cinvestav.mx/~grudsky/Papers/101.pdf

Albrecht Böttcher

Papers

Structured condition numbers of large Toeplitz matrices are rarely better than usual condition numbers

Toeplitz determinants with perturbations in the corners

Orthogonal Symmetric Toeplitz Matrices

Toeplitz operators on Hp

Pushing the envelope of the test functions in the Szego and Avram-Parter theorems