My question is: given a Fourier transform $\hat f$ of a function $f$, is it possible to estimate its asymptotic behaviour without performing the inverse transform?

Let me give a concrete example.

Suppose, a spherically symmetric real-valued function $f(x)$ (with $x\equiv|{\bf x}|$ and ${\bf x}\in \mathbb R^3$) has a Fourier transform of the following form $$ \hat f(k) = \frac{P_m(k)}{Q_n(k)} $$ where $P_m(k)$ and $Q_n(k)$ are certain polynomials of $k=|{\bf k}|$, and the degrees $m$ and $n$, some integer numbers less than 10. In the general case, it is quite hard (if at all possible) to calculate the inverse Fourier transform. However, suppose, we need not the function $f(x)$ itself, but only wish to determine its asymptotic behaviuor for $x\rightarrow \infty$. Such situations typically occur in physics, for instance in field theory, where the asymptotics of a field $f(x)$ is oftentimes significantly more relevant than the exact expression for this field.

So, can we extract any information about asymptotic behaviour of $f(x)$ form the Fourier image $\hat f(k)$ itself, without performing the complicated (and sometimes even technically impossible) inverse transform?

Thank you!