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Finite-Memory Systems

Authors:

Maria Alessandra Fasoli

Affiliation:

(1) Institut für Mathematik, Universität Innsbruck, Technikerstraße 25, A - 6020 Innsbruck, Austria

Abstract:

Let K be a field, k and n positive integers and let $A_1 , \ldots ,A_k {\text{ be }}n \times n -$ matrices with coefficients in K. For any function

$g\user2{: }\Gamma : = \left\{ {t: = (t_1 , \ldots ,t_k )} \right. \in \mathbb{N}^k \left| {\left. {t_1 t_2 \ldots t_k = 0} \right\}} \right. \to K^n$

there exists a unique solution $x:\mathbb{N}^k \to K^n$ of the system of difference equations

$\left( \right)x(t_1 + 1, \ldots ,t_k + 1) =$

$= \sum\nolimits_{j = 1}^k {A_j x} (t_1 + 1, \ldots ,t_{j - 1} + 1,t_j ,t_{j + 1} + 1, \ldots ,t_k + 1)$

defined by the matrix-k-tuple $(A_1 , \ldots ,A_k ) \in M(n;K)^k$ such that $x_{\left| \Gamma \right.} = g$ . The system $\left( \right)$ is called ldquo

finite-memory system rdquo

iff for every function g with finite support the values $x(t_1 , \ldots ,t_k )$ are 0 for sufficiently big $t_1 + \cdots + t_k$ . In the case $k = 2,K = \mathbb{R}$ , these systems and the corresponding matrix-k-tuples have been studied in bis, fm, fmv, fv1, fv, fz. In this paper I generalize these results to an arbitrary positive integer k and to an arbitrary field K.

Keywords:

system of difference equations finite-memory system nilpotent space of matrices separable k-tuples of matrices

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