I am interested in studying the steady state dynamics of a function, by means of looking at the Fourier transform. As a way to illustrate, suppose that the signal I want to study is the error function. In the problem I must solve, the function is slightly more complicated, but perhaps the intuition with the error function might help me enough.
We know that if $ f(t) = \frac{2}{\sqrt{\pi}} \int_0^t e^{-x^2}dx$, then,
$$ F(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(t) \cdot e^{i\cdot\omega\cdot t} dt = \frac{i}{\omega} e^{-\frac{\omega^2}{4}}\sqrt{\frac{2}{\pi}}.$$
Just based on the Fourier transform, I would like to be able to make conclusions about the long-term behavior $\lim_{t\to\infty}f(t)$. This is, because the function that I am studying does not seem to be easily invertible. Yet, we know in the case of the error function that:
$$\lim_{t \to \infty} \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt = 1$$
When only the Fourier transform is available, is there a way to look at the following limit:
$$ \lim_{t\to\infty}f(t) = \lim_{t\to\infty} \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty F(\omega) \cdot e^{-i\cdot\omega\cdot t} dw = ?? = 1 $$
(Disclosure: I asked a similar question before here Steady-state of a system and Fourier Transform, however, based on the comments I received, I thought it would be better to start with a new clearer, post. Thanks to Mattos and Maxim. I can delete the old one, if its necessary)