LEADER 05865nam 22007215 450 001 9910828902203321 005 20240516010838.0 010 $a1-4612-1426-2 024 7 $a10.1007/978-1-4612-1426-7 035 $a(CKB)3400000000089539 035 $a(SSID)ssj0001297219 035 $a(PQKBManifestationID)11858108 035 $a(PQKBTitleCode)TC0001297219 035 $a(PQKBWorkID)11362485 035 $a(PQKB)10731249 035 $a(DE-He213)978-1-4612-1426-7 035 $a(MiAaPQ)EBC3074002 035 $a(PPN)237995824 035 $a(EXLCZ)993400000000089539 100 $a20121227d1999 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aIntroduction to Large Truncated Toeplitz Matrices /$fby Albrecht Böttcher, Bernd Silbermann 205 $a1st ed. 1999. 210 1$aNew York, NY :$cSpringer New York :$cImprint: Springer,$d1999. 215 $a1 online resource (XI, 259 p.) 225 1 $aUniversitext,$x0172-5939 300 $a"With 62 figures." 311 $a0-387-98570-0 311 $a1-4612-7139-8 320 $aIncludes bibliographical references and index. 327 $a1 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. 330 $aIntroduction to Large Truncated Toeplitz Matrices is a text on the application of functional analysis and operator theory to some concrete asymptotic problems of linear algebra. The book contains results on the stability of projection methods, deals with asymptotic inverses and Moore-Penrose inversion of large Toeplitz matrices, and embarks on the asymptotic behavoir of the norms of inverses, the pseudospectra, the singular values, and the eigenvalues of large Toeplitz matrices. The approach is heavily based on Banach algebra techniques and nicely demonstrates the usefulness of C*-algebras and local principles in numerical analysis. The book includes classical topics as well as results obtained and methods developed only in the last few years. Though employing modern tools, the exposition is elementary and aims at pointing out the mathematical background behind some interesting phenomena one encounters when working with large Toeplitz matrices. The text is accessible to readers with basic knowledge in functional analysis. It is addressed to graduate students, teachers, and researchers with some inclination to concrete operator theory and should be of interest to everyone who has to deal with infinite matrices (Toeplitz or not) and their large truncations. 410 0$aUniversitext,$x0172-5939 606 $aTopology 606 $aMatrix theory 606 $aAlgebra 606 $aMathematical analysis 606 $aAnalysis (Mathematics) 606 $aTopology$3https://scigraph.springernature.com/ontologies/product-market-codes/M28000 606 $aLinear and Multilinear Algebras, Matrix Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M11094 606 $aAnalysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12007 615 0$aTopology. 615 0$aMatrix theory. 615 0$aAlgebra. 615 0$aMathematical analysis. 615 0$aAnalysis (Mathematics). 615 14$aTopology. 615 24$aLinear and Multilinear Algebras, Matrix Theory. 615 24$aAnalysis. 676 $a514 700 $aBöttcher$b Albrecht$4aut$4http://id.loc.gov/vocabulary/relators/aut$0345486 702 $aSilbermann$b Bernd$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910828902203321 996 $aIntroduction to large truncated Toeplitz matrices$9374408 997 $aUNINA