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

Philippe Ryckelynck (1) , Laurent Smoch (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

We study in this paper the functional equation \begin{center} $\displaystyle \alpha \mathbf{u}(t)+\mathcal{C}\star(\chi \mathbf{u})(t)=\mathbf{f}(t)$ \end{center} 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.

Mots clés: Functional equations; matricial polynomial functions; matrix sequences

Domaines: Optimisation et contrôle [math.OC]

Philippe Ryckelynck, Laurent Smoch. On the functional equation $\displaystyle \alpha \mathbf{u}+\mathcal{C}\star(\chi \mathbf{u})=\mathbf{f}$. North-Western European Journal of Mathematics, 2018, 4, pp.1-19. ⟨hal-00911185⟩