Can anyone help me evaluate the Fourier transform of of the following function, $t \in \mathbb{R}$, $\lambda \in \mathbb{C}$, $g:\mathbb{R} \rightarrow \mathbb{R}$,
$f(t) = \int_{t_0}^t e^{-\lambda(\tau - t)} g(\tau) d\tau$, for $t \geq t_0$.
This equation shows up in the state-space solution of a second-order linear initial value ODE. I'm trying to consolidate the state-space solution with one found using Fourier / Laplace transforms.
I have found a number of identities of Fourier transforms applied to infinite range convolutions. I'm not sure how or if they apply here. Any help would be appreciated.