The GLT class as a generalized Fourier analysis and applications

Linear Algebra and Its Applications

Volumn 419, Issue 1, 2006, Pages 180-233

  Serra Capizzano, Stefano ^a  

^a UNIVERSITY OF INSUBRIA   (Italy)

Author keywords

Algebra of sequences; Fourier Analysis; Generating function; Sparsely vanishing sequence or function; Toeplitz (and Generalized Locally Toeplitz) sequence

Indexed keywords

BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; FUNCTIONS; ITERATIVE METHODS; LINEAR ALGEBRA; MATHEMATICAL OPERATORS; PROBLEM SOLVING; THEOREM PROVING;

FOURIER ANALYSIS; GENERATING FUNCTIONS; SPARSELY VANISHING SEQUENCE OR FUNCTIONS; TOEPLITZ (AND GENERALIZED LOCALLY TOEPLITZ) SEQUENCES;

FOURIER TRANSFORMS;

EID: 33748294500     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2006.04.012     Document Type: Article

References (100)

1. In: Abramowitz M., and Stegun I.A. (Eds). Handbook of Mathematical Functions (1964), National Bureau of Standards, US Gov. Printing Office
2. Apel T. Anisotropic Finite Elements: Local Estimates and Applications (1999), Teubner, Stuttgart
3. Aricò A., Donatelli M., and Serra-Capizzano S. V-cycle optimal convergence for certain (multilevel) structured matrices. SIAM J. Matrix Anal. Appl. 26 1 (2004) 186-214
4. Avram F. On bilinear forms on Gaussian random variables and Toeplitz matrices. Probab. Theory Related Fields 79 (1988) 37-45
5. Axelsson O., and Barker V. Finite Element Solution of Boundary Value Problems, Theory and Computation (1984), Academic Press Inc., New York
6. Axelsson O., and Lindskög G. The rate of convergence of the preconditioned conjugate gradient method. Numer. Math. 52 (1986) 499-523
7. B. Beckermann, S. Serra-Capizzano, On the asymptotic spectrum of Finite Elements matrices, SIAM J. Numer. Anal., in press.
8. Bertaccini D., Golub G., Serra-Capizzano S., and Tablino Possio C. Preconditioned HSS method for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation. Numer. Math. 99 3 (2005) 441-484
9. Bertero M., and Boccacci P. Introduction to Inverse Problems in Imaging (1998), Inst. of Physics Publ., London
10. Bhatia R. Matrix Analysis (1997), Springer-Verlag, New York
11. Bhatia R. Pinching, trimming, truncating, and averaging of matrices. Amer. Math. Monthly 107 (2000) 602-608
12. Bini D., and Meini B. Approximate displacement rank and applications. In: Olshevsky V. (Ed). Contemp. Math. 281 (2001) 215-232
13. Böttcher A. The constants in the asymptotic formulas by Rambour and Seghier for inverses of Toeplitz matrices. Integral Equations Operator Theory 50 (2004) 43-55
14. Böttcher A. On the problem of testing the structure of a matrix by displacement operations. SIAM J. Numer. Anal. 44 (2006) 41-54
15. Böttcher A., and Silbermann B. Introduction to Large Truncated Toeplitz Matrices (1999), Springer-Verlag, New York
16. A. Böttcher, H. Widom, From Toeplitz eigenvalues through Green's Kernels to higher-order Wirtinger-Sobolev inequalities. Available from: .
17. Bronstein M., and Ivrii V. Sharp spectral asymptotics for operators with irregular coefficients. Pushing the limits. Comm. Partial Differential Equations 28 1/2 (2003) 83-102
18. Boutet de Monvel L., and Guillemin V. The Spectral Theory of Toeplitz Operators. Annals of Math. Stud. vol. 99 (1981), Princeton Univ. Press
19. Brandt A. Guide to multigrid development. In: Hackbusch W., and Trottenberg U. (Eds). Multigrid Methods. LNM vol. 960 (1982), Springer-Verlag, Berlin 220-312
20. Brezzi F., and Fortin M. Mixed and Hybrid Finite Element Methods (1991), Springer, New York
21. Chan R.H., and Chan T.F. Circulant preconditioners for elliptic problems. J. Numer. Linear Algebra Appl. 1 (1992) 77-101
22. Chan R.H., and Ng M. Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38 (1996) 427-482
23. Chan T.F., and Elman H.C. Fourier Analysis of iterative methods for Elliptic Problems. SIAM Rev. 31 1 (1989) 20-49
24. Ciarlet P. The Finite Element Method for Elliptic Problems (1978), North Holland, Amsterdam
25. Engl H., Hanke M., and Neubauer A. Regularization of Inverse Problems (1996), Kluwer Academic Publ., Dordrecht
26. Fiorentino G., and Serra-Capizzano S. Multigrid methods for Toeplitz matrices. Calcolo 28 (1991) 283-305
27. Fleckinger-Pellé J. The vibrations of a drum with fractal boundary. Mathematics and the 21st Century (Cairo, 2000) (2001), World Sci. Publishing, River Edge, NJ 305-322
28. Fleckinger J., and Métivier G. Théorie spectrale des opérateurs uniformément elliptiques sur quelques ouverts irréguliers. C.R. Acad. Sci. Paris Sér. A 276 (1973) 913-916
29. Gilkey P. Asymptotic Formulae in Spectral Geometry. Studies in Advanced Mathematics (2004), Chapman and Hall/CRC, Boca Raton, FL
30. L. Golinskii, S. Serra-Capizzano, The asymptotic properties of the spectrum of non symmetrically perturbed Jacobi matrix sequences, J. Approx. Theory, in press.
31. Golub G., and Melbo H. A stochastic approach to error estimates for iterative linear solvers: Part I. BIT 41 5 (2001) 977-985
32. Golub G.H., and Van Loan C.F. Matrix Computations (1983), The Johns Hopkins University Press, Baltimore
33. Gordon C., Webb D., and Wolpert S. Isospectral plane domains and surfaces via Riemannian orbifolds. Invent. Math. 110 1 (1992) 1-22
34. Greenbaum A. Analysis of a multigrid method as an iterative technique for solving linear systems. SIAM J. Numer. Anal. 21 3 (1984) 473-485
35. Grenander U., and Szego{combining double acute accent} G. Toeplitz Forms and Their Applications. second ed. (1984), Chelsea, New York
36. Grubb G. Functional Calculus of Pseudodifferential Boundary Problems. Progress in Mathematics (1996), Birkäuser, Basel
37. Gustafsson B., Kreiss H.O., and Oliger J. Time Dependent Problems and Difference Methods (1995), Wiley and Sons, New York
38. Hackbush W. Multigrid Methods and Applications (1979), Springer, New York
39. Heinig G., and Rost K. Algebraic Methods for Toeplitz-like Matrices and Operators (1984), Birkäuser, Basel
40. Hladnik M., Holbrook J., and Serra-Capizzano S. Fat diagonals and Fourier analysis. SIAM J. Matrix Anal. Appl. 24 4 (2003) 1060-1070
41. Holmgren S., and Otto K. Semicirculant solvers and boundary corrections for first-order partial differential equations. SIAM J. Sci. Comput. 17 3 (1996) 613-630
42. S. Holmgren, S. Serra-Capizzano, and P. Sundqvist, Can one hear the composition of a drum?, Mediterr. J. Math., in press.
43. S. Holmgren, S. Serra-Capizzano, P. Sundqvist, On the asymptotic spectrum of (non symmetric) Finite Difference matrix sequences, submitted for publication.
44. Hörmander L. Pseudo-differential operators and non-elliptic boundary problems. Ann. of Math. 2 83 (1966) 129-209
45. Horn R., and Johnson C. Topics in Matrix Analysis (1991), Cambridge University Press, Cambridge
46. T. Huckle, Multigrid preconditioning and Toeplitz matrices, Private communication, 1999.
47. Jackson D. The Theory of Approximation (1930), American Mathematical Society, New York
48. Jordan C. Cours d'Analyse de l'Ecole Polytecnique Vol. I (1909), Gauthier-Villars, Paris
49. Kac M. Can one hear the shape of a drum?. Amer. Math. Monthly 73 4 (1966) 1-23
50. Kac M., Murdoch W.L., and Szegö G. On the eigenvalues of certain Hermitian forms. J. Rational Mech. Anal. 2 (1953) 767-800
51. Kailath T., and Sayed A.H. Displacement structures: theory and applications. SIAM Rev. 37 (1995) 297-386
52. Kuijlaars A.B.J., and Serra-Capizzano S. Asymptotic zero distribution of orthogonal polynomials with discontinuously varying recurrence coefficients. J. Appr. Theory 113 (2001) 142-155
53. Lax P. On the stability of difference approximations to solutions of hyperbolic equations with variable coefficients. Comm. Pure Appl. Math. 14 (1961) 497-520
54. Lax P., and Wendroff B. On the stability of difference schemes. Comm. Pure Appl. Math. 15 (1962) 363-371
55. Lötstedt P. Convergence analysis of iterative methods by pseudodifference operators. J. Comput. Appl. Math. 115 1-2 (2000) 397-417
56. Manteuffel T., and Parter S. Preconditioning and boundary conditions. SIAM J. Numer. Anal. 27 3 (1990) 656-694
57. Miranda M., and Tilli P. On the asymptotical spectrum of Hermitian block Toeplitz matrices generated by matrix-valued functions. SIAM J. Matrix Anal. Appl. 21 3 (2000) 867-881
58. Nagy J., and O'Leary D. Restoring images degraded by spatially variant blur. SIAM J. Sci. Comput. 19 4 (1998) 1063-1082
59. Noutsos D., Serra-Capizzano S., and Vassalos P. Matrix algebra preconditioners for multilevel Toeplitz systems do not insure optimal convergence rate. Theoret. Computer Science 315 (2004) 557-579
60. Parter S.V. On the distribution of the singular values of Toeplitz matrices. Linear Algebra Appl. 80 (1986) 115-130
61. Parter S.V. On the eigenvalues of certain generalizations of Toeplitz matrices. Arch. Rat. Math. Mech. 3 (1962) 244-257
62. Potts D., Steidl G., and Nieslony A. Fast convolution with radial kernels at nonequispaced knots. Numer. Math. 98 2 (2004) 329-351
63. Rudin W. Real and Complex Analysis (1974), McGraw-Hill, New York
64. Ruge J.W., and Stüben K. Algebraic Multigrid. In: McCormick S. (Ed). Frontiers in Applied Mathematics: Multigrid Methods (1987), SIAM, Philadelphia 73-130
65. Seeley R. Complex powers of an elliptic operator. Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) (1966), AMS, Providence, RI 288-307
66. Serra-Capizzano S. An ergodic theorem for classes of preconditioned matrices. Linear Algebra Appl. 282 (1998) 161-183
67. Serra-Capizzano S. Asymptotic results on the spectra of block Toeplitz preconditioned matrices. SIAM J. Matrix Anal. Appl. 20 1 (1998) 31-44
68. Serra-Capizzano S. The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems. Numer. Math. 81 3 (1999) 461-495
69. Serra-Capizzano S. Spectral and computational analysis of block Toeplitz matrices with nonnegative definite generating functions. BIT 39 (1999) 152-175
70. Serra-Capizzano S. A note on the asymptotic spectra of finite difference discretizations of second order elliptic partial differential equations. Asian J. Math. 4 (2000) 499-514
71. Serra-Capizzano S. Korovkin tests, approximation, and ergodic theory. Math. Comput. 69 (2000) 1533-1558
72. Serra-Capizzano S. Distribution results on the algebra generated by Toeplitz sequences: a finite dimensional approach. Linear Algebra Appl. 328 (2001) 121-130
73. Serra-Capizzano S. Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences. Numer. Math. 92 3 (2002) 433-465
74. Serra-Capizzano S., and functions T. growth conditions and Toeplitz matrices. Rend. Circolo Mat. Palermo, Serie II 68 (2002) 791-795
75. Serra-Capizzano S. Generalized Locally Toeplitz sequences: spectral analysis and applications to discretized Partial Differential equations. Linear Algebra Appl. 366 1 (2003) 371-402
76. Serra-Capizzano S., Bertaccini D., and Golub G.H. How to deduce a proper eigenvalue cluster from a proper singular value cluster in the non normal case. SIAM J. Matrix Anal. Appl. 27 1 (2005) 82-86
77. S. Serra-Capizzano, P. Sundqvist, Stability of the notion of approximating class of sequences and applications, submitted for publication.
78. Serra-Capizzano S., and Tablino Possio C. Spectral and structural analysis of high precision Finite Difference matrices for Elliptic Operators. Linear Algebra Appl. 293 (1999) 85-131
79. Serra-Capizzano S., and Tablino Possio C. Positive representation formulas for finite difference discretizations of (elliptic) second order PDEs. In: Olshevsky V. (Ed). Contemp. Math. 281 (2001) 295-318
80. Serra-Capizzano S., and Tablino Possio C. Analysis of preconditioning strategies for collocation linear systems. Linear Algebra Appl. 369 (2003) 41-75
81. Serra-Capizzano S., and Tilli P. Extreme singular values and eigenvalues of non Hermitian block Toeplitz matrices. J. Comput. Appl. Math. 108 1/2 (1999) 113-130
82. Serra-Capizzano S., and Tilli P. On unitarily invariant norms of matrix valued linear positive operators. J. Inequalities Appl. 7 3 (2002) 309-330
83. Serra-Capizzano S., and Tyrtyshnikov E. How to prove that a preconditioner can not be superlinear. Math. Comput. 2 (2003) 1305-1316
84. Strikwerda J.C. Finite Difference Schemes and Partial Differential Equations (1989), Chapman and Hall, International Thompson Publ., New York
85. Tilli P. Locally Toeplitz matrices: spectral theory and applications. Linear Algebra Appl. 278 (1998) 91-120
86. Tilli P. A note on the spectral distribution of Toeplitz matrices. Linear Multilin. Algebra 45 (1998) 147-159
87. Tilli P. Singular values and eigenvalues of non-Hermitian block Toeplitz matrices. Linear Algebra Appl. 272 (1998) 59-89
88. Tilli P. Some results on complex Toeplitz eigenvalues. J. Math. Anal. Appl. 239 2 (1999) 390-401
89. Trefethen L.N., and Chapman S.J. Wave packet pseudomodes of twisted Toeplitz matrices. Comm. Pure Appl. Math. 57 9 (2004) 1233-1264
90. Trottenberg U., Oosterlee C.W., and Schüller A. Multigrid (2001), Academic Press, London
91. Tyrtyshnikov E. Influence of matrix operations on the distribution of eigenvalues and singular values of Toeplitz matrices. Linear Algebra Appl. 207 (1994) 225-249
92. Tyrtyshnikov E. A unifying approach to some old and new theorems on distribution and clustering. Linear Algebra Appl. 232 (1996) 1-43
93. Tyrtyshnikov E., and Zamarashkin N. Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships. Linear Algebra Appl. 270 (1998) 15-27
94. O. Widlund, MathSciNet revue of [23].
95. Widom H. Toeplitz matrices. In: Hirshman Jr. I. (Ed). Studies in Real and Complex Analysis (1965), Math. Ass. Amer.
96. Widom H. Asymptotic behaviour of block Toeplitz matrices and determinants. Adv. Math. 13 (1974) 284-322
97. Widom H. Szegö limit theorem: the higher dimensional matrix case. J. Funct. Analys. 39 (1980) 182-198
98. Widom H. Asymptotic Expansions of Pseudodifferential Operators on Bounded Domains. LNM vol. 1152 (1985), Springer-Verlag, New York
99. Wienands R., and Joppich W. Practical Fourier Analysis for Multigrid Methods (2004), CRC Press
100. Zygmund A. Trigonometric Series (1959), Cambridge University Press, Cambridge