Circulant recursive matrices

  • M. Barnabei
  • L. B. Montefusco


Recursive matrices are bi-infinite matrices which can be recursively generated starting from two given Laurent series α and β, called the recurrence rule and the boundary value of the matrix, respectively. The i th row of a recursive matrix contains the coefficients of the series α i β. These matrices were introduced by Barnabei, Brini, and Nicoletti [1] to formulate a more general version of the umbral calculus developed in a series of fundamental papers by Rota and his school [7, 8, 9]. In [2] it was shown that the most important operators of the umbral calculus can be represented by recursive matrices. In particular, shift-invariant operators, which play a crucial role in Rota’s theory, correspond to Toeplitz matrices, which can be characterized as recursive matrices with recurrence rule α(t) = t. Recently, recursive matrices were found to be useful in studying algebraic aspects of signal processing, since they contain the classes of bi-infinite Toeplitz, Hankel and Hurwitz matrices as special cases [3, 4].


Complex Polynomial Toeplitz Matrice Quotient Ring Circulant Matrix Circulant Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Barnabei, M., Brini, A., Nicoletti, G. (1982): Recursive matrices and umbral calculus. J. Algebra 75, 546–573MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Barnabei, M., Brini, A., Nicoletti, G. (1986): A general umbral calculus. In: Science and computers. Advances in Math. Suppl. Studies, 10. Academic Press, Orlando, FL, pp. 221–244Google Scholar
  3. [3]
    Barnabei, M., Guerrini, C., Montefusco, L.B. (1998): Some algebraic aspects of signal processing. Linear Alg. Appl. 284, 3–17MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Barnabei, M., Montefusco, L.B. (1998): Recursive properties of Toeplitz and Hurwitz matrices. Linear Alg. Appl. 274, 367–388MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Davis, P.J. (1994): Circulant matrices. 2nd edition. Chelsea, New YorkGoogle Scholar
  6. [6]
    Gantmacher, F.R. (1959): The theory of matrices, vols. 1, 2. Chelsea, New YorkGoogle Scholar
  7. [7]
    Mullin, R., Rota, G.-C. (1970): On the foundations of combinatorial theory. III. Theory of binomial enumeration. In: Harris, B. (ed.), Graph theory and its applications. Academic Press, New York pp. 167–213Google Scholar
  8. [8]
    Roman, S.M., Rota, G.-C. (1978): The umbral calculus. Adv. Math. 27, 95–188MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Rota, G.-C., Kahaner, D., Odlyzko, A. (1973): On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, 684–760MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2001

Authors and Affiliations

  • M. Barnabei
  • L. B. Montefusco

There are no affiliations available

Personalised recommendations