On the functional equation $\displaystyle \alpha\bf{u}+\mathcal{C}\star(\chi \bf{u})=\bf{f}$

TL;DR

This paper investigates a matrix-valued functional equation combining recurrence and delay elements, establishing conditions for solutions using new results on linear independence in matrix algebras.

Contribution

It introduces a matricial framework for solving a class of functional equations involving convolution and delay, with new insights into linear independence of matrix monomials.

Findings

Established existence and uniqueness conditions for solutions.

Developed a new approach using linear independence of matrix monomials.

Provided a framework bridging recurrence and delayed functional equations.

Abstract

We study in this paper the functional equation $\displaystyle \alpha \mathbf{u}(t)+\mathcal{C}\star(\chi \mathbf{u})(t)=\mathbf{f}(t)$ where $\alpha\in\mathbb{C}^{d\times d}$, $\mathbf{u},\mathbf{f}:\mathbb{R}\rightarrow\mathbb{C}^d$, $\mathbf{u}$ being unknown. The term $\mathcal{C}\star(\chi \mathbf{u})(t)$ denotes the discrete convolution of an almost zero matricial mapping $\mathcal{C}$ with discrete support together with the product of $\mathbf{u}$ and the characteristic function $\chi$ of a fixed segment. This equation combines some aspects of recurrence equations and/or delayed functional equations, so that we may construct a matricial based framework to solve it. We investigate existence, unicity and determination of the solution to this equation. In order to do this, we use some new results about linear independency of monomial words in matrix algebras.