Introduction to Large Truncated Toeplitz Matrices

Bernd Silbermann

ISBN 10: 0387985700 / ISBN 13: 9780387985701

Editore: Springer New York, Springer New York Nov 1998, 1998

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Neuware -Introduction to Large Truncated Toeplitz Matrices is a text on the application of functional analysis and operator theory to some concrete asymptotic problems of linear algebra. The book contains results on the stability of projection methods, deals with asymptotic inverses and Moore-Penrose inversion of large Toeplitz matrices, and embarks on the asymptotic behavoir of the norms of inverses, the pseudospectra, the singular values, and the eigenvalues of large Toeplitz matrices. The approach is heavily based on Banach algebra techniques and nicely demonstrates the usefulness of C\*-algebras and local principles in numerical analysis. The book includes classical topics as well as results obtained and methods developed only in the last few years. Though employing modern tools, the exposition is elementary and aims at pointing out the mathematical background behind some interesting phenomena one encounters when working with large Toeplitz matrices. The text is accessible to readers with basic knowledge in functional analysis. It is addressed to graduate students, teachers, and researchers with some inclination to concrete operator theory and should be of interest to everyone who has to deal with infinite matrices (Toeplitz or not) and their large truncations.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 276 pp. Englisch. Codice articolo 9780387985701

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Titolo: Introduction to Large Truncated Toeplitz ...

Casa editrice: Springer New York, Springer New York Nov 1998

Data di pubblicazione: 1998

Legatura: Buch

Condizione: Neu

Descrizione articolo

Riassunto:

Applying functional analysis and operator theory to some concrete asymptotic problems of linear algebra, this book contains results on the stability of projection methods, deals with asymptotic inverses and Moore-Penrose inversion of large Toeplitz matrices, and embarks on the asymptotic behaviour of the norms of inverses, the pseudospectra, the singular values, and the eigenvalues of large Toeplitz matrices. The approach is heavily based on Banach algebra techniques and nicely demonstrates the usefulness of C*-algebras and local principles in numerical analysis, including classical topics as well as results and methods from the last few years. Though employing modern tools, the exposition is elementary and points out the mathematical background behind some interesting phenomena encountered with large Toeplitz matrices. Accessible to readers with basic knowledge in functional analysis, the book addresses graduates, teachers, and researchers and should be of interest to everyone who has to deal with infinite matrices (Toeplitz or not) and their large truncations.

Contenuti:

1 Infinite Matrices.- 1.1 Boundedness and Invertibility.- 1.2 Laurent Matrices.- 1.3 Toeplitz Matrices.- 1.4 Hankel Matrices.- 1.5 Wiener-Hopf Factorization.- 1.6 Continuous Symbols.- 1.7 Locally Sectorial Symbols.- 1.8 Discontinuous Symbols.- 2 Finite Section Method and Stability.- 2.1 Approximation Methods.- 2.2 Continuous Symbols.- 2.3 Asymptotic Inverses.- 2.4 The Gohberg-Feldman Approach.- 2.5 Algebraization of Stability.- 2.6 Local Principles.- 2.7 Localization of Stability.- 3 Norms of Inverses and Pseudospectra.- 3.1C*-Algebras.- 3.2 Continuous Symbols.- 3.3 Piecewise Continuous Symbols.- 3.4 Norm of the Resolvent.- 3.5 Limits of Pseudospectra.- 3.6 Pseudospectra of Infinite Toeplitz Matrices.- 4 Moore-Penrose Inverses and Singular Values.- 4.1 Singular Values of Matrices.- 4.2 The Lowest Singular Value.- 4.3 The Splitting Phenomenon.- 4.4 Upper Singular Values.- 4.5 Moler’s Phenomenon.- 4.6 Limiting Sets of Singular Values.- 4.7 The Moore-Penrose Inverse.- 4.8 Asymptotic Moore-Penrose Inversion.- 4.9 Moore-Penrose Sequences.- 4.10 Exact Moore-Penrose Sequences.- 4.11 Regularization and Kato Numbers.- 5 Determinants and Eigenvalues.- 5.1 The Strong Szegö Limit Theorem.- 5.2 Ising Model and Onsager Formula.- 5.3 Second-Order Trace Formulas.- 5.4 The First Szegö Limit Theorem.- 5.5 Hermitian Toeplitz Matrices.- 5.6 The Avram-Parter Theorem.- 5.7 The Algebraic Approach to Trace Formulas.- 5.8 Toeplitz Band Matrices.- 5.9 Rational Symbols.- 5.10 Continuous Symbols.- 5.11 Fisher-Hartwig Determinants.- 5.12 Piecewise Continuous Symbols.- 6 Block Toeplitz Matrices.- 6.1 Infinite Matrices.- 6.2 Finite Section Method and Stability.- 6.3 Norms of Inverses and Pseudospectra.- 6.4 Distribution of Singular Values.- 6.5 Asymptotic Moore-Penrose Inversion.- 6.6 Trace Formulas.- 6.7 The Szegö-Widom Limit Theorem.- 6.8 Rational Matrix Symbols.- 6.9 Multilevel Toeplitz Matrices.- 7 Banach Space Phenomena.- 7.1 Boundedness.- 7.2 Fredholmness and Invertibility.- 7.3 Continuous Symbols.- 7.4 Piecewise Continuous Symbols.- 7.5 Loss of Symmetry.- References.- Symbol Index.