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Author Bรถttcher, Albrecht. author Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis [electronic resource] / by Albrecht Bรถttcher, Sergei M. Grudsky Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2000 http://dx.doi.org/10.1007/978-3-0348-8395-5 X, 112 p. online resource

SUMMARY

The subject of this text is the relation between the properties of infinite Toeplitz matrices ao a_I a_2 al ao a_I a2 al ao and their large finite sections This is very big and even inexhaustible subject, and therefore we must limit ourselves to a few concrete problems here. We will focus our attention on singular values. The singular values of An are the eigenvalues of (AÃn)I/2. The properties of the singular values of An for fixed n (or, as in so-called interlacing theorems, for some consecutive n) are studied in linear algebra. The problem of determining the singular values of An for large n (say n = 700) is a business of numerical linear algebra. The behavior of the singular 23 values of An for n --+ 00 (or, say, for n = 10 ) is a concern of asymptotic linear algebra. Finally, the investigation of the properties of the infinite matrix A is a task of functional analysis. To get an idea of what this text is about, we cite a few questions we will consider. Preface viii Question 1. Does the smallest singular value 81 (An) stay away from zero as n -t oo? Because this is the question whether the norms IIA;;111 are uniformly bounded for all sufficiently large n

CONTENT

1 Infinite Toeplitz Matrices -- 2 C*-Algebras in Action -- 3 Instability -- 4 Condition Numbers -- 5 Singular Values -- Notation Index

Mathematics Mathematical analysis Analysis (Mathematics) Mathematics Analysis

Location

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