Applying Laguerre's function for approximate calculation of Green's function of a second-order differential equation

We consider the equation $\ddot x(t)=Ax(t)+f(t)$, $t\in\mathbb{R}$, with the matrix coefficient $A$. This equation has a unique solution $x$, which is bounded on $\mathbb{R}$, for any continuous bounded inhomogeneity $f$ if and only if the spectrum of the matrix $A$ does not intersect the semi-axis $\mathbb{R}_-=\{z\in\mathbb{R}: z\le0\}$. In this case, the solution $x$ is defined by the formula
\begin{equation*} x(t)=\int_{-\infty}^{+\infty}G(t-s)f(s)\,ds, \quad G(t)=-\frac12 e^{-\sqrt{A}|t|}(\sqrt{A})^{-1}. \end{equation*}
We discuss the problem of approximate calculation of Green's function $G(t)$ using its expansion into Laguerre's series. The scale parameter $\tau$ in Laguerre's polynomials is chosen to ensure the highest accuracy.