Block Locally Toeplitz Sequences: Construction and Properties

  • Carlo Garoni
  • Stefano Serra-Capizzano
  • Debora SesanaEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 30)


The theory of block locally Toeplitz (LT) sequences—along with its generalization known as the theory of block generalized locally Toeplitz (GLT) sequences—is a powerful apparatus for computing the spectral distribution of matrices arising from the discretization of differential problems. In this paper we develop the theory of block LT sequences, whereas the theory of block GLT sequences is the subject of the complementary paper (Chap. 3 of this book).


Singular values and eigenvalues Block locally Toeplitz sequences Block Toeplitz matrices Discretization of differential equations 



Carlo Garoni is a Marie-Curie fellow of the Italian INdAM under grant agreement PCOFUND-GA-2012-600198. The work of the authors has been supported by the INdAM GNCS (Gruppo Nazionale per il Calcolo Scientifico). The authors wish to thank Giovanni Barbarino for useful discussions.


  1. 1.
    Avram, F.: On bilinear forms in Gaussian random variables and Toeplitz matrices. Probab. Theory Relat. Fields 79, 37–45 (1988)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barbarino, G.: Equivalence between GLT sequences and measurable functions. Linear Algebra Appl. 529, 397–412 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bhatia, R.: Matrix Analysis. Springer, New York (1997)CrossRefGoogle Scholar
  4. 4.
    Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Springer, New York (1999)CrossRefGoogle Scholar
  5. 5.
    Böttcher, A., Silbermann B.: Analysis of Toeplitz Operators, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  6. 6.
    Donatelli, M., Garoni, C., Mazza, M., Serra-Capizzano, S., Sesana, D.: Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol. Appl. Math. Comput. 245, 158–173 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications (Volume I). Springer, Cham (2017)CrossRefGoogle Scholar
  8. 8.
    Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications (Volume II). Springer, Cham (2018)CrossRefGoogle Scholar
  9. 9.
    Garoni, C., Serra-Capizzano, S., Sesana, D.: Spectral analysis and spectral symbol of d-variate \(\mathbb Q_{\boldsymbol {p}}\) Lagrangian FEM stiffness matrices. SIAM J. Matrix Anal. Appl. 36, 1100–1128 (2015)Google Scholar
  10. 10.
    Garoni, C., Serra-Capizzano, S., Vassalos, P.: A general tool for determining the asymptotic spectral distribution of Hermitian matrix-sequences. Oper. Matrices 9, 549–561 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Garoni, C., Mazza, M., Serra-Capizzano, S.: Block generalized locally Toeplitz sequences: from the theory to the applications. Axioms 7, 49 (2018)CrossRefGoogle Scholar
  12. 12.
    Garoni, C., Serra-Capizzano, S., Sesana, D.: Block generalized locally Toeplitz sequences: topological construction, spectral distribution results, and star-algebra structure. In: Bini, D.A., et al. (eds.) Structured Matrices in Numerical Linear Algebra. Springer INdAM Series, vol. 30, pp. 59–79. Springer, Cham (2019). Google Scholar
  13. 13.
    Garoni, C., Speleers, H., Ekström, S.-E., Reali, A., Serra-Capizzano, S., Hughes, T.J.R.: Symbol-based analysis of finite element and isogeometric B-spline discretizations of eigenvalue problems: exposition and review. Arch. Comput. Meth. Eng. (in press).
  14. 14.
    Grenander, U., Szegő, G.: Toeplitz Forms and Their Applications, 2nd edn. AMS Chelsea Publishing, New York (1984)Google Scholar
  15. 15.
    Parter, S.V.: On the distribution of the singular values of Toeplitz matrices. Linear Algebra Appl. 80, 115–130 (1986)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Serra-Capizzano, S.: Distribution results on the algebra generated by Toeplitz sequences: a finite dimensional approach. Linear Algebra Appl. 328, 121–130 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Serra-Capizzano, S.: More inequalities and asymptotics for matrix valued linear positive operators: the noncommutative case. Oper. Theory Adv. Appl. 135, 293–315 (2002)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Serra-Capizzano, S.: Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations. Linear Algebra Appl. 366, 371–402 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Serra-Capizzano, S.: The GLT class as a generalized Fourier analysis and applications. Linear Algebra Appl. 419, 180–233 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Serra-Capizzano, S., Tilli, P.: On unitarily invariant norms of matrix-valued linear positive operators. J. Inequal. Appl. 7, 309–330 (2002)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tilli, P.: A note on the spectral distribution of Toeplitz matrices. Linear Multilinear Algebra 45, 147–159 (1998)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tilli, P.: Locally Toeplitz sequences: spectral properties and applications. Linear Algebra Appl. 278, 91–120 (1998)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tyrtyshnikov, E.E.: A unifying approach to some old and new theorems on distribution and clustering. Linear Algebra Appl. 232, 1–43 (1996)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Tyrtyshnikov, E.E., Zamarashkin, N.L.: Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships. Linear Algebra Appl. 270, 15–27 (1998)MathSciNetCrossRefGoogle Scholar
  25. 25.
    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, 1191–1201 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Carlo Garoni
    • 1
    • 2
  • Stefano Serra-Capizzano
    • 2
    • 3
  • Debora Sesana
    • 2
    Email author
  1. 1.University of Italian SwitzerlandInstitute of Computational ScienceLuganoSwitzerland
  2. 2.University of InsubriaDepartment of Science and High TechnologyComoItaly
  3. 3.Uppsala UniversityDepartment of Information Technology, Division of Scientific ComputingUppsalaSweden

Personalised recommendations