On the functional equation αu + C ⋆ (χu) = f

Abstract

We study in this paper the functional equation \[ \alpha \mathbf{u}(t)+\mathcal{C}\star(\chi \mathbf{u})(t)=\mathbf{f}(t) \] 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 independence of monomial words in matrix algebras.