On the functional equation $\displaystyle α\bf{u}+\mathcal{C}\star(χ\bf{u})=\bf{f}$
Year: 2015
Type: preprint
Abstract: We study in this paper the functional equation $\displaystyle α\mathbf{u}(t)+\mathcal{C}\star(χ\mathbf{u})(t)=\mathbf{f}(t)$ where $α\in\mathbb{C}^{d\times d}$, $\mathbf{u},\mathbf{f}:\mathbb{R}\rightarrow\mathbb{C}^d$, $\mathbf{u}$ being unknown. The term $\mathcal{C}\star(χ\mathbf{u})(t)$ 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 $χ$ 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.
Source: arXiv (Cornell University)
Authors: Ryckelynck, Philippe, Smoch, Laurent
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4294343501