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Böttcher A., Grudsky S.M. — Spectral Properties of Banded Toeplitz Matrices
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Название: Spectral Properties of Banded Toeplitz Matrices
Авторы: Böttcher A., Grudsky S.M.
Аннотация: This self-contained introduction to the behavior of several spectral characteristics of large Toeplitz band matrices is the first systematic presentation of a relatively large body of knowledge. Covering everything from classic results to the most recent developments, Spectral Properties of Banded Toeplitz Matrices is an important resource. The spectral characteristics include determinants, eigenvalues and eigenvectors, pseudospectra and pseudomodes, singular values, norms, and condition numbers. Toeplitz matrices emerge in many applications and the literature on them is immense. They remain an active field of research with many facets, and the material on banded ones until now has primarily been found in research papers. The book may serve both as a text for introducing the material and as a reference. The approach is based on the know-how and experience of the authors in combining functional analytical methods with hard analysis and in applying operator theoretical methods to matrix theory, which reveals the essence of several phenomena and leads to significant improvements in existing results. All basic results presented in the book are precisely stated as theorems and accompanied by full proofs. Audience This book is written for applied mathematicians, engineers, and scientists who encounter Toeplitz matrices in their research. It also will be of interest to mathematicians in the fields of operator theory, numerical analysis, structured matrices, or random matrix theory, and physicists, chemists, biologists, and economists who deal with stationary statistical and stochastic problems. Parts of the book are suitable for use as a graduate-level text on Toeplitz matrices or analysis.
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
ed2k: ed2k stats
Год издания: 2005
Количество страниц: 411
Добавлена в каталог: 17.04.2010
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Предметный указатель
211
250 261
, 219
60
33
347
43
335
347
43
, 6
179
3
11
347 355
178
28
90
11
3
79
313
223
, probability 46 61 377
49 61 377
181
96
9
347
157
157
157
341
313
31
165
219
2
64
6
4
90
315
328
313
313
313
315
315
137
262
262
262
complex numbers
natural numbers
real numbers
, complex unit circle 3
integers
nonnegative integers
9
59
211
, 167
9
59
249
8
8
, 8 85
8 309
8 183
101
223
, boundary 31
315
313
, 212
335
variance
211
225
211
59
122
101
96
5
90
60
225
249
12 60
Algebra 240
Algebra, 240
Algebra, Banach 240
Algebra, F lner 377
Algebra, irrational rotation 378
Algebra, Wiener 2
Algorithm, fast 77
Algorithm, superfast 77
Anderson model 374
Approximation number 211
Asymptotically extended sequence 302
Asymptotically good pseudoeigenvalue 300
Asymptotically good pseudomode 300
Asymptotically localized sequence 302 305
Avram — Parter theorem 219
Banach algebra 240
Banach algebra, unital 240
Banach — Steinhaus theorem 60
Baxter — Gohberg — Feldman theorem 63
Baxter — Schmidt formula 37
Bergman space 75
Beta distribution 233
Bounded variation 219
Branch point 264 367
Brown — Halmos theorem 102
Cauchy singular integral operator 75
Cauchy’s interlacing theorem 224
Chebyshev polynomial 22
CIRC 32
Circulant matrix 32
Cluster 224
Coker A 9
Cokernel 9
Componentwise condition number 332
Condition number 137
Condition number, componentwise 332
Condition number, for matrix inversion 328
Condition number, full structured 313
Condition number, normwise 313
Condition number, structured 313
Confluent Vandermonde 42
Convergence, critical behavior 177
Convergence, strong 59
Convergence, uniform 59
Convergence, weak 59
Critical transient phase 177
Determinant 46
Discrete Hamiltonian 335
Discrete Laplacian 335
Duduchava — Roch formula 92
Eigenvalue density 379
Essential spectrum 9
Exp , exp 7
Expectation 225
Expected value 347
Exponentially decaying sequence 15
Extended sequence 17
F lner algebra 377
Factorization, Wiener — Hopf 7
Fast algorithm 77
Fej r mean 122
Field of values 167
Finite section method 64
Formula, Aaxter — Schmidt 37
Formula, Duduchava — Roch 92
Formula, Gohberg — Sementsul 77
Formula, Trench’s 41
Formula, Widom’s 38 65
Fourier coeffcients 2
Fourier matrix 32
Fredholm operator 9
Function of bounded variation 219
Galerkin method 74
Gauss — Seidel iteration 187
Gohberg — Sementsul formula 77
Hadamard’s inequality 80
Hamiltonian, discrete 335
Hankel matrix 2
Hardy space 11
Hardy’s inequality 91
Hatano — Nelson model 375
Higher order relative spectrum 175
Hilbert — Schmidt operator 45
Hirschman’s theorem 274
Homomorphism of -algebras 241
Hull, polynomial convex 208
Hull, polynomial numerical 179
Hurwitz’ theorem 355
Im A 9
Image 9
Ind A 9
INDEX 9
Inequality, Hadamard’s 80
Inequality, Hardy’s 91
Instability index 155
Interval matrix 256
Inverse closedness 240
Involution 240
Irrational rotation -algebra 378
Jacobi’s theorem 48
Ker A 9
Kernel 9
Krein — Rutman theorem 253
Kreiss matrix theorem 181
Laplacian, discrete 335
Laurent matrix 28
Laurent polynomial 8
Lim inf 163
Lim sup 163
Lin linear hull
LOG natural logarithm
Matrix, circulant 32
Matrix, finite Toeplitz 31
Matrix, Fourier 32
Matrix, infinite Hankel 2
Matrix, infinite Toeplitz 1
Matrix, Laurent 28
Matrix, monodromy group 367
Matrix, positive definite 101
Matrix, positive semi-definite 101
Matrix, Toeplitz-like 123
Matrix, tridiagonal Toeplitz 34
Norm surface 178
Normal solvability 9
Normwise condition number 313
Nowhere locally constant 364
Numerical range 167
Operator, Cauchy singular integral 75
Operator, Fredholm 9
Operator, Hilbert — Schmidt 45
Operator, normally solvable 9
Operator, trace class 45
Operator, Wiener — Hopf 74
Order of a zero 88
Polynomial convex hull 208
Polynomial numerical hull 179
Positive definite 101
Positive semi-definite 101
Preconditioning 259
Proper cluster 224
Pseudoeigenvalue 300
Pseudoeigenvalue, asymptotically good 300
Pseudomode 300
Pseudomode, asymptotically good 300
Pseudospectrum 157
Pseudospectrum, structured 157
Rad 12
Resolution of the identity 21
Resolvent 9
Riesz — Markov theorem 377
Schmidt — Spitzer theorem 274
Second order relative spectrum 175
Sequence, asymptotically extended 302
Sequence, asymptotically localized 302 305
Sequence, exponentially decaying 15
Sequence, extended 17
Sequence, stable 61
Singular value 211
Singular value decomposition 212
Singular value interlacing 216
Sky region 178
Sp 9
Space, Bergman 75
Space, Hardy 11
Spectral distribution 377
Spectral radius 12
Spectrum 9
Spectrum, absolutely continuous 21
Spectrum, essential 9
Spectrum, higher order relative 175
Spectrum, point 21
Spectrum, second order relative 175
Spectrum, singular continuous 21
Splitting phenomenon 212
Stable sequence 61
Stone’s formula 21
Strong convergence 59
Structured condition number 313
Structured condition number, for matrix inversion 328
Structured condition number, full 313
Structured pseudospectrum 157
Superfast algorithm 77
Symbol 3
Szeg — Widom limit theorem 47
Szeg ’s strong limit theorem 44
Theorem, Avram — Parter 219
Theorem, Banach — Steinhaus 60
Theorem, Baxter — Gohberg — Feldman 63
Theorem, Brown — Halmos 102
Theorem, Cauchy’s interlacing 224
Theorem, Hirschman’s 274
Theorem, Hurwitz 355
Theorem, Jacobi’s 48
Theorem, Krein — Rutman 253
Theorem, Kreiss matrix 181
Theorem, Riesz — Markov 377
Theorem, Schmidt — Spitzer 274
Theorem, singular value interlacing 216
Theorem, Szeg — Widom limit 47
Theorem, Szeg ’s strong limit 44
Theorem, Wiener’s 6
Toeplitz matrix 1 31
Toeplitz-like matrix 123
Tr 248
Trace 248
Trace class operator 45
Tracial state 377
Trench’s formula 41
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