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Böttcher A., Grudsky S.M. — Spectral Properties of Banded Toeplitz Matrices
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.


Язык: en

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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$a_{j}^{(p)}(A)$      211
$a_{\varrho}$      250 261
$BV$, $BV[a, b]$      219
$c_{0}$      60
$C_{n}(b)$      33
$d_{jk}(K)$      347
$E(a)$      43
$E_{jj}$      335
$E_{jk}$      347
$G(a)$      43
$GW$, $GW_{\pm}$      6
$G_{k}(A_{n})$      179
$H(a)$      3
$H^{2}$      11
$H_{\Omega}^{jk}(b)$      347 355
$J_{n}(\lambda)$      178
$L(A)$      28
$l^{2}(\beta)$      90
$L^{p} := L^{p}(\mathbf{T})$      11
$l^{p}:=l^{p}(\mathbf{Z}_{+})$      3
$l^{p}_{n}$      79
$M_{n}(\mathbf{K})$      313
$N_{n}(E)$      223
$P$, probability $P_{n}$      46 61 377
$Q_{n}$      49 61 377
$R_{n}(\lambda)$      181
$S(b)$      96
$sp_{ess} A$      9
$sp_{\Omega}^{(j,k)} A$      347
$sp_{\varepsilon} A$      157
$sp_{\varepsilon}^{(p)} A$      157
$sp_{\varepsilon}^{B,C} A$      157
$sp_{\varepsilon}^{m} A$      341
$Str_{n}(\mathbf{K})$      313
$T_{n}(a)$      31
$V(a)$      165
$V_{[a,b]}f$      219
$W$      2
$W_{n}$      64
$W_{\pm}$      6
$\chi_{n}$      4
$\eta_{\beta}$      90
$\kappa(A, x)$      315
$\kappa^{Str}(A)$      328
$\kappa^{Str}(A, x)$      313
$\kappa^{Str}_{b}(A, x)$      313
$\kappa^{Str}_{full}(A, x)$      313
$\kappa_{b}(À, x)$      315
$\kappa_{full}(A, x)$      315
$\kappa_{p}(A_{n})$      137
$\Lambda(b)$      262
$\Lambda_{s}(b)$      262
$\Lambda_{w}(b)$      262
$\mathbf{c}$      complex numbers
$\mathbf{N}$      natural numbers
$\mathbf{R}$      real numbers
$\mathbf{T}$, complex unit circle $T(a)$      3
$\mathbf{Z}$      integers
$\mathbf{Z}_{+}$      nonnegative integers
$\mathcal{B}(X)$      9
$\mathcal{B}(X, Y)$      59
$\mathcal{F}_{j}^{(n)}$      211
$\mathcal{H}_{X}(A)$,$\mathcal{H}_{p}(A)$      167
$\mathcal{K}(X)$      9
$\mathcal{K}(X, Y)$      59
$\mathcal{O}$      249
$\mathcal{P}$      8
$\mathcal{P}^{+}$      8
$\mathcal{P}^{+}_{s}$, $\mathcal{P}^{+}_{n}$      8 85
$\mathcal{P}_{r,s}$      8 309
$\mathcal{P}_{r}$      8 183
$\mathcal{R}(A)$      101
$\mu_{n}(E), \mu(E)$      223
$\partial$, boundary $D_{n}(a)$      31
$\Phi(A,x)$      315
$\Phi^{Str}(A,x)$      313
$\Sigma(A)$, $\Sigma_{p}(A)$      212
$\sigma(T)$      335
$\sigma^{2}$      variance
$\sigma_{j} (A)$      211
$\Sigma_{J}$      225
$\sigma_{j}^{(p)}(A)$      211
$\sigma_{min}(A)$      59
$\sigma_{n}b$      122
$\sim$      101
$\simeq$      96
$\tilde{a}$      5
$\xi_{\beta}$      90
$\||\cdot\||_{p}$      60
$\||\cdot\||_{p}_{F}$      225
$\||\cdot\||_{p}_{tr}$      249
$\||\cdot\||_{p}_{\infty}$      12 60
Algebra      240
Algebra, $C^{*}$      240
Algebra, Banach      240
Algebra, F${\o}$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 $W$, exp $W_{\pm}$      7
Expectation      225
Expected value $E_{j}$      347
Exponentially decaying sequence      15
Extended sequence      17
F${\o}$lner algebra      377
Factorization, Wiener — Hopf      7
Fast algorithm      77
Fej$\acute{e}$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 $C^{*}$-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 $C^{*}$-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 $M_{n}$      163
Lim sup $M_{n}$      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 $A$      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 $A$      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$\ddot{o}$ — Widom limit theorem      47
Szeg$\ddot{o}$’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$\ddot{o}$ — Widom limit      47
Theorem, Szeg$\ddot{o}$’s strong limit      44
Theorem, Wiener’s      6
Toeplitz matrix      1 31
Toeplitz-like matrix      123
Tr $A$      248
Trace      248
Trace class operator      45
Tracial state      377
Trench’s formula      41
1 2
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