Circulant recursive matrices

• M. Barnabei
• L. B. Montefusco
Chapter

Abstract

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].

Keywords

Complex Polynomial Toeplitz Matrice Quotient Ring Circulant Matrix Circulant Matrice

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