Suppose I have a function $f(t) \in L^2(\mathcal{R})$ and it is specified by:
$f(t) = \int_{-\infty}^{\infty} H(\omega) \exp(-\beta t \omega) \exp(i \omega t)\,d\omega$
Suppose $H(\omega)\in L^2(\mathcal{R})$, and we know it is among the subset of $L^2(\mathcal{R})$ which is Fourier transformable,but we don't know anything more specific than that. Is it possible to find an analytical formula for $\hat{f}(\omega)$, the Fourier transform of $f(t)$, in terms of $H(\omega)$ and $\beta$?