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 


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  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

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© Springer-Verlag Italia 2001

Authors and Affiliations

  • M. Barnabei
  • L. B. Montefusco

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