Given the differential equation
\begin{align} \frac{d v}{d t} = - v(t) + \kappa * v \end{align}
where $\kappa$ is some linear temporal filter (like a sum of two exponentials, for instance) and $*$ denotes temporal convolution. Moreover, given we know the fixpoint, can we check its stability simply by taking the Fourier transformation and looking at the real part of the derivative with respect to $\hat v$ at every Fourier frequency?
I.e. can I do the following?
\begin{align} F[\frac{d v}{d t}] = F[- v + \kappa * v] \end{align}
\begin{align} \leftrightarrow \frac{d \hat v}{d t} = - \hat v(n) + \hat \kappa(n) \hat v(n) \end{align}
And then simply look at the real part of $d \hat f/d \hat v$ with $\hat f = - \hat v + \hat \kappa \hat v$? Thus, the fixpoint is stable if $- 1 + Re (\hat \kappa (n)) < 0$ for every Fourier frequency $n$?
Thanks, and sorry for any inconvenience regarding the layout of my question. I only have access to my phone right now, which makes typing formulas difficult. Any refactoring appreciated.