Fourier: low frequency for long time expansion

Fourier: low frequency for long time expansion

pI wanted to know if a certain result on Fourier transform is true. Let's
suppose to have a function $f(x)$ and his Fourier transform $\tilde f(q)$.
We now that, by applying Fourier inversion, we can recover $f(x)$ from
$f(q)$ . Let's consider now a new funcion $g(q)$ such that $f(q) \sim
g(q)$ for $q \rightarrow 0$. By Fourier inversion we can obtain the new
function in real space $g(x)$. When is it true that $f(x) \sim g(x)$ for
$x \rightarrow \infty$? I think this is intuitive reasonable for my
comprehension. Nevertheless my considerations till now bring to the
conclusion that this is normally false. This is becaues I know that if the
function $f(q)$ analitically continued to complex space has some poles on
the lower plane then the function $f(x)$ decays esponentially with a power
law given by the position of the pole. So if we have two different
functions behaving in the same way as $q \rightarrow 0$ but with different
poles, their behaviour at infinity will be different. Is this dicussion
correct? But what to say then about the intuitive result??? Thanks in
advance for any answer!/p