References Dorostkar A., Neytcheva M., Serra-Capizzano S.Spectral analysis of coupled PDEs

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4 302 References 62. Garoni C., Serra-Capizzano S. The theory of generalized locally Toeplitz sequences: a review, an extension, and a few representative applications. Oper. Theory. Adv. Appl. 259 (2017) Garoni C., Serra-Capizzano S. Spectral distribution results beyond the algebra generated by variable-coefficient Toeplitz sequences: the GLT approach. J. Fourier Anal. Appl. (in press). DOI link: Garoni C., Serra-Capizzano S. Generalized Locally Toeplitz Sequences: Theory and Applications (Volume II). In preparation for Springer. 65. Garoni C., Serra-Capizzano S., Sesana D. Tools for determining the asymptotic spectral distribution of non-hermitian perturbations of Hermitian matrix-sequences and applications. Integr. Equ. Oper. Theory 81 (2015) Garoni C., Serra-Capizzano S., Sesana D. Spectral analysis and spectral symbol of d-variate Q p Lagrangian FEM stiffness matrices. SIAM J. Matrix Anal. Appl. 36 (2015) Garoni C., Serra-Capizzano S., Vassalos P.A general tool for determining the asymptotic spectral distribution of Hermitian matrix-sequences. Oper. Matrices 9 (2015) Golinskii L., Serra-Capizzano S.The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences. J. Approx. Theory 144 (2007) Golub G. H., Van Loan C. F. Matrix Computations. Fourth Edition, The Johns Hopkins University Press, Baltimore (2013). 70. Grenander U., Szegő G. Toeplitz Forms and Their Applications. Second Edition, AMS Chelsea Publishing, New York (1984). 71. Higham N. J. Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008). 72. Holbrook J. No dice: a deterministic approach to the Cartan centroid. J. Ramanujan Math. Soc. 27 (2012) Hörmander L. Pseudo-differential operators and non-elliptic boundary problems. Annals of Math. 83 (1966) Hughes T. J. R., Cottrell J. A., Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg. 194 (2005) Hughes T. J. R., Evans J. A., Reali A. Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput. Methods Appl. Mech. Engrg. 272 (2014) Hughes T. J. R., Reali A., Sangalli G. Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p-method finite elements with k-method NURBS. Comput. Methods Appl. Mech. Engrg. 197 (2008) Hughes T. J. R., Reali A., Sangalli G. Efficient quadrature for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Engrg. 199 (2010) Iannazzo B. The geometric mean of two matrices from a computational viewpoint. Numer. Linear Algebra Appl. 23 (2016) Jeuris B., Vandebril R., Vandereycken B. A survey and comparison of contemporary algorithms for computing the matrix geometric mean. Electron. Trans. Numer. Anal. 39 (2012) Kelley J. L. General Topology. Van Nostrand (1955). 81. Kuijlaars A. B. J., Serra-Capizzano S. Asymptotic zero distribution of orthogonal polynomials with discontinuously varying recurrence coefficients. J. Approx. Theory 113 (2001) Kuijlaars A. B. J., Van Assche W. The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients. J. Approx. Theory 99 (1999) Lapuyade-Lahorgue J., Barbaresco F. Radar detection using Siegel distance between autoregressive processes, application to HF and X-band radar. Radar Conference, RADAR 08. IEEE (2008) 1 6.

5 References Lawson J., Lim Y. Monotonic properties of the least squares mean. Math. Ann. 351 (2011) Mascarenhas H., Silbermann B. Sequences of variable-coefficient Toeplitz matrices and their singular values. J. Funct. Anal. 270 (2016) Miranda M., Tilli P. Asymptotic spectra of Hermitian block Toeplitz matrices and preconditioning results. SIAM J. Matrix Anal. Appl. 21 (2000) Nakamura N. Geometric means of positive operators. Kyungpook Math. J. 49 (2009) Parter S. V. On the extreme eigenvalues of Toeplitz matrices. Trans. Amer. Math. Soc. 100 (1961) Parter S. V. On the distribution of the singular values of Toeplitz matrices. Linear Algebra Appl. 80 (1986) Pinkus A. Totally positive matrices. Cambridge University Press (2010). 91. Quarteroni A. Numerical models for differential problems. Second Edition, Springer- Verlag Italia, Milan (2014). 92. Reali A. An isogeometric analysis approach for the study of structural vibrations. J. Earthquake Engrg. 10 (2006) Reali A., Hughes T. J. R. An introduction to isogeometric collocation methods. Chapter 4 of the Springer Book Isogeometric Methods for Numerical Simulation, Edited by G. Beer and S. Bordas, CISM, Udine (2015). 94. Roman F., Manni C., Speleers H. Spectral analysis of matrices in Galerkin methods based on generalized B-splines with high smoothness. Numer. Math. 135 (2017) Royden H. L., Fitzpatrick P. M. Real Analysis. Fourth Edition, Pearson Education Asia Limited and China Machine Press (2010). 96. Rudin W. Principles of Mathematical Analysis. Third Edition, McGraw-Hill, New York (1976). 97. Rudin W. Real and Complex Analysis. Third Edition, McGraw-Hill, Singapore (1987). 98. Salinelli E., Serra-Capizzano S., Sesana D. Eigenvalue-eigenvector structure of Schoenmakers Coffey matrices via Toeplitz technology and applications. Linear Algebra Appl. 491 (2016) Schillinger D., Evans J. A., Reali A., Scott M. A., Hughes T. J. R. Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Comput. Methods Appl. Mech. Engrg. 267 (2013) Shargorodsky E. Toeplitz matrices with variable coefficients, pseudodifferential operators, and Strichartz s method. Math. Nachr. 283 (2010) Serra-Capizzano S. On the extreme spectral properties of Toeplitz matrices generated by L 1 functions with several minima/maxima. BIT 36 (1996) Serra-Capizzano S. On the extreme eigenvalues of Hermitian (block) Toeplitz matrices. Linear Algebra Appl. 270 (1998) Serra-Capizzano S. How bad can positive definite Toeplitz matrices be? Numer. Funct. Anal. Optimiz. 21 (2000) Serra-Capizzano S. Distribution results on the algebra generated by Toeplitz sequences: a finite dimensional approach. Linear Algebra Appl. 328 (2001) Serra-Capizzano S. More inequalities and asymptotics for matrix valued linear positive operators: the noncommutative case. Oper. Theory Adv. Appl. 135 (2002) Serra-Capizzano S. Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations. Linear Algebra Appl. 366 (2003) Serra-Capizzano S. The GLT class as a generalized Fourier analysis and applications. Linear Algebra Appl. 419 (2006) Serra-Capizzano S., Bertaccini B., Golub G. H. How to deduce a proper eigenvalue cluster from a proper singular value cluster in the nonnormal case. SIAM J. Matrix Anal. Appl. 27 (2005) Serra-Capizzano S., Sesana D. Approximating classes of sequences: the Hermitian case. Linear Algebra Appl. 434 (2011)

6 304 References 110. Serra-Capizzano S., Sesana D., Strouse E. The eigenvalue distribution of products of Toeplitz matrices Clustering and attraction. Linear Algebra Appl. 432 (2010) Serra-Capizzano S., Sundqvist P. Stability of the notion of approximating class of sequences and applications. J. Comput. Appl. Math. 219 (2008) Serra-Capizzano S., Tablino-Possio C. Analysis of preconditioning strategies for collocation linear systems. Linear Algebra Appl. 369 (2003) Serra-Capizzano S., Tilli P. On unitarily invariant norms of matrix-valued linear positive operators. J. Inequal. Appl. 7 (2002) Sesana D. Spectral distributions of structured matrix-sequences: tools and applications. Ph.D. Thesis in Mathematics of Computation, University of Insubria, Como, Italy (2010). Full text available at: Silbermann B., Zabroda O. Asymptotic behavior of generalized convolutions: an algebraic approach. J. Integral Equ. Appl. 18 (2006) Simonenko I. B. Szegő-type limit theorems for generalized discrete convolution operators. Math. Notes 78 (2005) Smith G. D. Numerical Solution of Partial Differential Equations: Finite Difference Methods. Third Edition, Oxford University Press, New York (1985) Speleers H. Inner products of box splines and their derivatives. BIT Numer. Math. 55 (2015) Tilli P. A note on the spectral distribution of Toeplitz matrices. Linear and Multilinear Algebra 45 (1998) Tilli P. Locally Toeplitz sequences: spectral properties and applications. Linear Algebra Appl. 278 (1998) Tilli P. Some results on complex Toeplitz eigenvalues. J. Math. Anal. Appl. 239 (1999) Tilli P. Universal bounds on the convergence rate of extreme Toeplitz eigenvalues. Linear Algebra Appl. 366 (2003) Tyrtyshnikov E. E. A unifying approach to some old and new theorems on distribution and clustering. Linear Algebra Appl. 232 (1996) Tyrtyshnikov E. E., Zamarashkin N. L. Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships. Linear Algebra Appl. 270 (1998) Tyrtyshnikov E. E., Zamarashkin N. L. A general equidistribution theorem for the roots of orthogonal polynomials. Linear Algebra Appl. 366 (2003) Widom H. Extreme eigenvalues of translation kernels. Trans. Amer. Math. Soc. 100 (1961) Zabroda O. N. Generalized convolution operators and asymptotic spectral theory. Dissertation, Department of Mathematics, TU Chemnitz, Zabroda O. N., Simonenko I. B. Asymptotic invertibility and the collective asymptotic spectral behavior of generalized one-dimensional discrete convolutions. Funct. Anal. Appl. 38 (2004) Zamarashkin N. L., Tyrtyshnikov E. E. Distribution of eigenvalues and singular values of Toeplitz matrices under weakened conditions on the generating function. Sb. Math. 188 (1997)

7 Index Symbols A n (a), 175 {A n } n GLT κ, 143 {A n } n LT a f, 120 {A n } n λ f, 46 {A n } n λ φ, 46 {A n } n σ f, 46, 47 {A n } n σ φ, 45 {A n } n σ, λ f, 46 a.c.s. {B n,ε } n { A n } n, 93 a.c.s. {B n,m } n { A n } n, 71 C, 179 C [0,1], 121 C c (C), 8 C c (D), 13 Cc m(r), 8 C c (R), 8 C(D), 14 C m n, 7 C m (R), 8 C n ( f ), 109 C[z], 58 χ E, 9 d a.c.s. ({A n } n, {B n } n ), 70 d measure ( f, g), 22 D n (a), 119 D n [p] (v), 236 D(S,ε), 8 D(z,ε), 8 E, 10 ER( f ), 10 ess inf D f = ess inf x D f (x), 11 ess sup D f = ess sup x D f (x), 11 f (A), 41 f k, 96 F n, 107 φ g (F), 10 φ γ (F) = F(γ ), 16 φ [q], 233 G, 149 G, 156 g( f ) = g f, 9 H 1 ( ), 218 H 1 0 ( ), 219 i, 8 I m, I, 7 I(X), 8 L 1, 175 L p (D), 9, 12 LTn m (a, f ), 119 λ j (X), 8 λ max (X), 8 λ min (X), 8 (X), 8 M, 9 M D, 9 m f, M f, 99 μ k, 9 f L p, f L p (D), 9, 12 g, 9 g,d, 9 x p, 8, 27 X p, 8, 28 x, 8, 28 X, 8, 28 X p, 8, 31 O m, O, 7 p a.c.s. ({A n } n ), 68, 71 p measure ( f ), 22 P n, 183 R +, 254 R(X), 8 R m n, 7 Springer International Publishing AG 2017 C. Garoni and S. Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications, DOI /

8 306 Index ρ(x), 8 #S, 7 S, 7 S n (a), 190 something t τ something else, 8 supp( f ), 13 σ j (X), 8 σ max (X), 8 σ min (X), 8 T, 173 T n ( f ), 96 τ a.c.s., 66, 70 τ d, 21 τ measure, 22, 24 u v, 119 V sp. C m, 34 w 1 w d, 8 ω g (δ), 9 X, 8, 31 x T, x, 7 X T, X, 7 X Y, 7 X Y, 291 X Y, X > Y, 7 ζ n = o(ξ n ), 8 ζ n = O(ξ n ), 8 A A.e., 9 Addition of matrix-sequences, 54 Algebra generated by Toeplitz sequences T, 173 generated by variable-coefficient Toeplitz sequences C, 179 of GLT pairs G, 156, 163 of GLT sequences G, 156 of matrix-sequences E, 66, 163 of measurable functions M, 156, 163 of zero-distributed sequences Z, 53 product algebra E M, 156, 163 Algebraic-topological definitions of GLT sequences, 163 ALM axioms, 183 Analytical predictions of the eigenvalue errors, 259 Anisotropic, 217 Approximating class of sequences (a.c.s.), 66 as ε 0, 92, 93 formed by Hermitian matrices, 79 Approximation space, 230, 244, 254 Approximation theory for matrix-sequences, 65, 74 Arrow-shaped sampling matrix, 190 Associativity of tensor products, 40 Asymptotic of λ min (T n ( f )), 113 Attraction, 50 Automatic procedure for computing symbols, 262 Avram Parter theorem, 108 Axelsson Lindskog estimates, 65 B Bandwidth, 97 Barycenter, 184 Bilinearity of tensor products, 40 of the LT operator, 121 Birth of LT sequences, 115, 193 Block GLT sequence, 261 Block Toeplitz matrix, 261 Boundary conditions Dirichlet, 198 Dirichlet Neumann, 210 Neumann, 202 Bounded metric, 68 B-splines, 229, 231, 232 Building blocks of the theory of GLT sequences, 52, 95, 125 C CAD, 229 Calculus, 231, 245, 255 Cardinal B-spline, 233 Cardinality, 7 Cauchy interlacing theorem, 289 Cauchy Schwarz inequality, 12 Central B-spline basis functions, 234 Central Greville abscissae, 235 Cesàro means, 288 CG, 3 Characterization of a.c.s. parameterized by ε 0, 93 of concave functions, 26 of GLT sequences, 143, 151 of LT sequences, 137 of Riemann-integrable functions, 19 of Schatten p-norms, 32 of s.u. matrix-sequences, 84 of s.v. matrix-sequences, 160 of zero-distributed sequences, 52 Chebyshev inequality, 14 Circulant matrix, 106, 109

9 Index 307 Class of sequences, 92 Closeness a.c.s., 73 in measure, 73 in norm, 73 of functions, 73 of matrices, 65, 72 Closure, 7, 11, 51 of GLT pairs, 149 of GLT sequences, 149 of σ -pairs, 82 Clustering, 49 Coarse grid, 218 Collocation matrix, 230 Collocation method, 230 Collocation points, 230 Complex polynomial, 58 Computational cost, 218, 229 Condition number, 112 Congruence invariance, 183 Conjugate exponents, 12 Conjugate transpose, 7 Conjugate transposition of a GLT sequence, 155 of an a.c.s., 83 of an LT sequence, 136 of matrix-sequences, 54 Consistency with scalars, 183 Continuous transition, 118 Convection matrix, 252 Convection term, 199, 202 Convention, 36, 54, 71, 242 Convergence a.c.s., 66, 70 a.e., 14 dominated, 15 in a pseudometric space, 21 in L p, 13 in measure, 14, 22 superlinear (of CG), 3 uniform, 157 Countable, 275 Counting measure, 73 Criteria to identify a.c.s., 89 D De l Hôpital theorem, 113 Decomposition of a Green matrix, 187 Degree of a trigonometric polynomial, 97 Density in L p, 13 in the space of GLT pairs, 151, 163 in the space of GLT sequences, 150 DEs, 1 Diagonal part, 187 Diagonal sampling matrix, 119 Differential calculus, 231, 245, 255 Differential operator associated with a DE, 2 higher-order, 195, 224, 241, 252, 262 in non-divergence form, 204 nonnegative, 196, 210, 212, 242 translation-invariant, 116 Diffusion matrix, 252 Diffusion model, 218 Diffusion term, 199 Direct sum, 40, 120 Dirichlet boundary conditions, 198 Dirichlet Neumann boundary conditions, 210 Discretization parameter, 2 Disk, 8, 21 Distance (pseudometric), 20 Divergence form, 198 Dominated convergence theorem, 15 E Eckart Young theorem, 30 Eigenpairs, 255 Eigenspaces, 41 Eigenvalue errors, 255, 259 Elliptic problem, 192 Engineers, 257, 261 Equivalence relation, 21, 22 Equivalent pseudometrics, 22 inducing τ measure, 25 Essential infimum, 11 Essential range, 10, 50, 111 Essential supremum, 11 Essential zero, 113 Evaluation functional, 16 Exponential of a matrix, 43 Exponential series, 271 ε-expansion, 8, 49 F FD, 3 FD formula, 195, 196, 210, 212 FE, 3 Fine grid, 218 Formal structure of the symbol, 195, 224, 241, 252 Fourier analysis, 4 Fourier coefficients, 96, 175

10 308 Index Fourier frequency, 17 Fourier series, 13, 175 Fourier sum, 178 Fourier transform, 107 Fourier variable, 4, 195, 224, 242 Fractional DEs, 3 Function bounded, 9 characteristic (indicator), 9 concave, 25 continuous a.e., 5, 19, 152 convex, 27 holomorphic (analytic), 59 measurable, 9 Riemann-integrable, 19, 127 subadditive, 27 tensor-product, 8, 119 Function of a matrix, 41 Functional, 10 evaluation, 16 G Galerkin method, 244, 254 Galerkin problem, 220, 226, 244, 255 Generalized convolution, 176 Generalized eigenvalue problem, 255 Generalized Locally Toeplitz (GLT) sequence, 143, 170 Generating function of a sequence of diagonal sampling matrices, 119 of a Toeplitz sequence, 96 of a variable-coefficient Toeplitz sequence, 175 of an LT sequence, 118, 121 Generator of circulant matrices, 106 Geometric arithmetic mean inequality, 101 Geometric mean of matrices, 183, 184 Geometry map (function), 230, 245, 255, 262 Gershgorin circle theorem, 284 G-images, 217 GLT, 1 GLT algebra, 156 GLT analysis of FD discretization matrices, 193, 200, 203, 204, 208, 211, 213 of FE discretization matrices, 220, 226 of IE discretization matrices, 187 of IgA discretization matrices, 240, 249, 256 GLT pairs, 149 Gradual transition, 118 Green matrix, 187 Greville abscissae, 232, 235 Grid that accumulates at a point, 217 H Hadamard product, 7 Hat-functions, 219, 225, 234 Heine Cantor theorem, 16 Hermitian matrix-sequence, 165 Higher-order differential operator, 195, 224, 241, 252, 262 Hölder inequality, 12 Hölder-type inequality, 31 Hörmander theory, 195 HPD, 7 HPSD, 7 Hyperrectangle, 13, 48 Hypersurface, 47 I IEs, 1 IgA, 3 IgA collocation matrices, 235 IgA collocation method, 229 IgA Galerkin method, 229, 244, 254 Imaginary part of a function, 8, 19 of a matrix, 8, 38, 62 Imaginary unit, 8 Incremental ratio, 26 Informal meaning of singular value distribution, 46 of spectral (eigenvalue) distribution, 46 Integral expression, 98 Interior, 11, 60, 111 Interlacing theorem for eigenvalues, 36, 81 for singular values, 36, 76 Isogeometric Analysis (IgA), 229 Isogeometric collocation approximation, 229 Isogeometric collocation method, 229 Isogeometric Galerkin approximation, 244, 254 Isogeometric Galerkin method, 229, 244, 254 Isoparametric approach, 231 Iterative methods (solvers), 3

11 Index 309 J Jacobi matrix, 110, 176 Jensen inequality, 12 Jordan normal form, 268 K Karcher mean of matrices, 184 Kernel (of an IE), 185 Kernel (symbol), 121, 143, 170 Ky-Fan theorem, 38 L Leading principal submatrix, 118, 194 Least squares problem, 31 Lebesgue integral, 19 Lebesgue measure, 9 Lebesgue s characterization theorem of Riemann-integrable functions, 19 Linear combination of a.c.s., 83 of GLT sequences, 145, 155 of LT sequences, 136 of matrix-sequences, 171 Linear FEs, 218, 225 Linear operator, 221, 250, 294, 296 Linear positive operator (LPO), 100, 101 Localization of the spectrum, 35, 98 Locally Toeplitz (LT) operator, 119 Locally Toeplitz (LT) sequence, 118, 120 Local method, 3, 222 Local refinement, 217, 254 Local support property, 232 Lower-order differential operators, derivatives, terms, 196, 199, 200, 207, 210, 236, 246 Lower triangular part, 187 LPO, 100 LT, 1 Lusin theorem, 15 M Mapping of a uniform grid (mesh), 212, 213, 254 Mass matrix, 255 Matrix 2-level Toeplitz, 185 arrow-shaped sampling, 190 banded, 96 block Toeplitz with Toeplitz blocks, 185 circulant, 106, 109 collocation, 230 diagonal sampling, 119, 236 Green, 187 Jacobi, 110, 176 mass, 255 normal, 29, 30 shift-invariant, 116 skew-hermitian, 29 skew-symmetric, 226 stiffness, 220, 226, 245, 255 Toeplitz, 96 unitarily diagonalizable, 29 variable-coefficient Toeplitz, 175 Matrix function, 41 Matrix-sequence, 10 Hermitian, 165 sparsely unbounded (s.u.), 84 sparsely vanishing (s.v.), 159 strongly clustered, 49 strongly clustered (in the sense of the eigenvalues), 49 strongly clustered in the sense of the singular values, 49 weakly clustered, 49 weakly clustered (in the sense of the eigenvalues), 49 weakly clustered in the sense of the singular values, 49 Mergelyan theorem, 59 Metric, 68, 72 Metric space, 20, 22 Microscope, 119 Minimax principle for eigenvalues, 34 for singular values, 34 Minimizer, 184 Minimum norm vector, 31 Mixed FD/FE technique, 225 Modulus of continuity, 9 Monotone (operator), 100 Moore Penrose pseudoinverse, 8, 31 of a GLT sequence, 161 of an a.c.s., 88 Multigrid, 3 Multilevel GLT sequence, 261 Multivariate GLT sequence, 261 N Natural operations on functions, 156 on matrix-sequences, 54 on pairs in E M, 156 Neumann boundary conditions, 202

12 310 Index Nodal values, 186, 192, 213 Non-divergence form, 204 Nonnegativity of the symbol, 196, 210, 212, 242 Non-uniform grid, 212, 254 Norm 1-, 28 2-, 28 -, 28 Frobenius, 33 L p -, 9, 12 nuclear, 8 operator, 8, 27 p-, 8, 27 Schatten 1-, 33 Schatten 2-, 33 Schatten -, 33 Schatten p-, 8, 31 spectral (Euclidean), 8, 28, 29 trace-, 8 unitarily invariant, 28, 31, 32, 101 Notation from probability theory, 10 Numerical eigenvalues, 255 NURBS, 229, 231, 257 Perturbation, 4, 57, 116, 166, 202 Perturbation theorem for eigenvalues, 35, 81 for singular values, 35, 76 Weyl s, 35 Physical variable, 4, 195 Pinching inequality, 32 Polynomial set, 59 Preconditioned CG, 65 Preconditioned Krylov methods, 3 Preconditioned matrices, 162 Principal submatrix, 118 Principal symbol, 195 Product componentwise (Hadamard), 7 of a.c.s., 87 of GLT sequences, 155 of LT sequences, 136 of matrix-sequences, 54 tensor (Kronecker), 40, 120 Pseudometric, 20, 22, 27, 66, 70, 82, 149 Pseudometric space, 20 O Operations on functions, 156 on matrix-sequences, 54 on pairs in E M, 156 Operations ops, 154 Order of a differential operator, 196, 210, 212, 242 Order of an essential zero, 113 Order of the zero at θ = 0, 196, 210, 212, 242 Orthogonal polynomials, 111, 176 Orthogonal projection, 32 Orthogonality relations, 96 Orthonormal bases, 32, 101, 102 Oscillatory, 217 Outliers, 47, 65, 243, 259 P Paradigm, 229 Parametric (reference) domain, 230, 245, 255 Parter theorem, 113 Partition, 19, 153 Partition of unity, 233 Periodic extension, 114 Permutation invariance, 183 Q Quadrature, 229 Quotient space, 21, 22 R Radar, 185 Radius, 8, 21, 60, 62 Rank of a function, 73, 74 Reaction matrix, 252 Reaction term, 199, 200 Real part of a function, 8, 19 of a matrix, 8, 38 Rearranged version, 47, 48, 196, 197, 200, 224, 227, 228, 243, 258 Rectangle formula, 186 Reduced GLT sequence, 262 Reference (parametric) domain, 230, 245, 255 Refinement, 217, 254 Regular map, 217, 249, 253 Residual term, 236 Restriction, 113 Riemann integral, 19 Riemann sum, 46, 110 Riemannian distance, 184

13 Index 311 S S.u., 84 S.v., 159 Saddle point form, 225 Saddle point structure, 226 Scalar-multiplication of matrix-sequences, 54 Schur complement, 225, 226 Schur normal form, 268 Sequence of diagonal sampling matrices, 127, 152 Sequence of matrices, 10 Set closed, 11, 59, 82, 149, 150, 163 compact, 13, 60 connected, 59, 60, 111 dense, 13, 151, 163 measurable, 9 of GLT pairs, 149, 150, 156, 163 of σ -pairs, 82 open, 21 polynomial, 59 Sherman Morrison Woodbury formula, 65 Singular map, 217, 254 Singular value decomposition (SVD), 29 Singular value distribution, 45 of a finite sum of LT sequences, 133 of a GLT sequence, 144 of FD discretization matrices, 193, 200, 203, 205, 208, 211, 214 of FE discretization matrices, 220, 225, 227 of IE discretization matrices, 187 of IgA discretization matrices, 240, 249, 253, 256 of matrix-sequences beyond C, 180 of matrix-sequences beyond T, 174 of matrix-sequences in C, 180 of matrix-sequences in T, 174 of matrix-sequences perturbed by zero-distributed sequences, 82 of preconditioned matrices, 162 of the geometric (Karcher) mean of GLT sequences, of Toeplitz sequences, 108 Singular values, 30 Singularity point, 217 Sobolev space, 218 Sobolev (weak) derivative, 218 Space of matrix-sequences E, 10 Space of measurable functions M, 9 Space of measurable functions M D, 9 Sparsely unbounded (s.u.) matrix-sequence, 84 Sparsely vanishing (s.v.) matrix-sequence, 159 SPD, 7 Spectral (eigenvalue) distribution, 46 of a finite sum of LT sequences, 135 of a GLT sequence, 145, 146 of FD discretization matrices, 193, 200, 203, 205, 208, 211, 214 of FE discretization matrices, 220, 225, 227 of Hermitian matrix-sequences perturbed by zero-distributed sequences, 83 of IE discretization matrices, 187 of IgA discretization matrices, 240, 249, 253, 256 of matrix-sequences beyond C, 180 of matrix-sequences beyond T, 174 of matrix-sequences in C, 180 of matrix-sequences in T, 174 of preconditioned matrices, 162 of the geometric (Karcher) mean of GLT sequences, of Toeplitz sequences, 108, 111 Spectral attraction, 50 Spectral decomposition of a matrix function, 42 of circulant matrices, 107 Spectral radius, 8 Splitting, 66, 79, 89, 157 SPSD, 7 Standard differential calculus, 231, 245, 255 Stiffness matrix, 220, 226, 245, 255 Strip, 176 Strong attraction with infinite order, 50 Sturm Liouville problem, 117, 192 Submultiplicative property, 28 Support, 13 SVD, 29 Symbol, 120, 143, 170 principal, 195 singular value, 46 spectral (eigenvalue), 46 Symmetric approximation of a matrix, 205, 208, 240 Szegő first limit theorem, 108, 174 Szegő formulas, 116 weighted, 117 σ -pair, 82

14 312 Index T Tensor (Kronecker) product, 40, 120 Tilli class, 111, 174 Tilli theorem, 111, 114 Toeplitz matrix, 96 Toeplitz sequence, 96, 130 Tool, 57, 65, 74, 79 Topological basis, 11 Topological interpretation, 82, 149 Topologically equivalent pseudometrics, 22 inducing τ measure, 25 Topology, 20, 68, 72 a.c.s. (τ a.c.s. ), 66, 70, 72, 82, 149, 150, 163 of convergence in measure (τ measure ), 22, 24, 72, 82, 149, 150, 163 product, 22, 82, 149, 150, 163 pseudometric, 21 pseudometrizable, 21 Trace-norm inequalities, 33 Transformed convection coefficient, 231 Transformed diffusion coefficient, 231 Transformed problem, 231, 241, 252 Transformed reaction coefficient, 231 Translation invariance, 116 Transpose, 7 Trigonometric monomial, 13, 103, 151 Trigonometric polynomial, 13, 17, 100, 137, 191, 195, 224, 241 Truncation, 152 U Uniform (equispaced) grid, 46, 116, 118, 212 Uniform (equispaced) samples, 46, 48, 118 Uniform knots, 235 Uniform knot sequence, 232, 233 Uniformly shifted and scaled versions, 234 Unilevel GLT sequence, 261 Uniqueness of the symbol of a GLT sequence, 145, 170 of the symbol of an LT sequence, 134 Univariate GLT sequence, 261 Upper triangular part, 187 Urysohn s lemma, 50 V Vanishing property, 12 Vanishment on the boundary, 233 Variable-coefficient Toeplitz matrix, 175 Variable-coefficient Toeplitz sequence, 175 Variational characterization of Schatten p-norms, 32 Vector of eigenvalues, 196, 198, 201 Vector of samples, 196, 198, 201 W Way of reasoning, 263 Weak form, 219, 225, 244, 254 Weak hypotheses on DE coefficients, 218, 225 Weak (Sobolev) derivative, 218 Weierstrass theorem, 157, 279 Weight function, 117, 118, 121 Weighted Szegő formulas, 117 Well-posedness, 192 Weyl s majorization theorem, 34 Weyl s perturbation theorem, 35 Wiener-type condition, 176 Z Zero-distributed sequence, 52, 70, 82, 125, 185 Zero of exponential order, 114 Zone (of a point), 22