Algebraic Framework for Discrete Dynamical Systems over Laurent Series

TL;DR

This paper extends the algebraic framework for discrete dynamical systems to include Laurent series, enabling modeling of bidirectional systems with applications in data processing and control.

Contribution

It generalizes existing algebraic frameworks to Laurent polynomials and series, addressing open questions and enhancing modeling capabilities for multidimensional systems.

Findings

Preserves duality and adjoint properties in the extended framework

Provides a rigorous mathematical foundation with proofs and examples

Demonstrates applications in data processing and control theory

Abstract

We generalize the framework of discrete algebraic dynamical systems \cite{Andriamifidisoa2014} to Laurent polynomials and series over \(\Z^r\), enabling the modeling of bidirectional discrete systems. By redefining the spaces \(\Dprime\) and \(\Aprime\), introducing a bilinear mapping (defined as the scalar product in Section 3), and extending the shift operator, we preserve the duality and adjoint properties of \cite{Andriamifidisoa2014}. These properties are rigorously proved and illustrated through examples and a data processing case study on bidirectional sequence transformations. In contrast to Oberst \cite{Ob90}, our algebraic approach emphasizes the structure of Laurent series, providing a streamlined framework for multidimensional systems. This work addresses an open question from \cite{Andriamifidisoa2014} and has applications in multidimensional data processing, such as image filtering and control theory.