Discrete linear Algebraic Dynamical Systems

TL;DR

This paper explores the duality between multi-indexed sequences and finite support sequences, using scalar products to define orthogonals, and connects polynomial operators with shift operators in discrete linear dynamical systems.

Contribution

It introduces a duality framework for sequence spaces and relates polynomial shift operators to polynomial multiplication, generalizing to vector sequences and power series.

Findings

Established duality between sequence spaces using scalar products

Connected polynomial shift operators with polynomial multiplication as adjoints

Generalized results to vector sequences and power series

Abstract

The vector space of the multi-indexed sequences over a field and the vector space of the sequences with finite support are dual to each other, with respect to a \textit{scalar product}, which we used to define \textit{orthogonals} in these spaces. The closed subspaces in the first vector space are then the orthogonals of subsets in the second space. Using power series and polynomials, we prove that the \textit{polynomial operator in the shift} which U. Oberst and J. C. Willems have introduced to define time invariant discrete linear dynamical systems is the functorial adjoint of the polynomial multiplication. These results are generalized to the case of vectors of sequences and vectors of power series and polynomials. We end this paper by describing discrete linear algebraic dynamical systems.