Is there a way to integrate the following so that $f(t,z)$ does not lose its dependence on $t$? (Both $f(t,z)$ and $g(t')$ are integrable functions; $r$ is some constant.)
$f(t,z)=\int^{+\infty}_{-\infty}dk \int^t_0dt'g(t') \text{e}^{ik(z+rt')}$
I've tried integrating over $k$ first, then obtaining a $\delta$ function and finally integrating over $t'$. However, this gives $f(t,z)$ as a function of $z$ only:
$$\int^{+\infty}_{-\infty}dk \int^t_0dt'g(t') \text{e}^{ik(z+rt')}=2\pi\int^t_0dt'g(t') \dfrac{\delta(z/r+t')}{r}=\dfrac{2\pi g(-z/r)}{r}$$
Edit: $f(t)$ amended to $f(t,z)$; $r$ defined to be just some constant.