For a function $f(x)$ with Fourier transform $\hat{F}(q)$, I'm interested in understanding the relationship of the Fourier transform of a power of a derivative of $f$ to $\hat{F}(q)$. Explicitly, I want to express
$$\mathcal{F}[f](q\,;n,k) = \int_{-\infty}^{\infty}dx\,\left(\frac{d^nf}{dx^n}\right)^ke^{-iqx}$$
in terms of $\hat{F}(q)$.
I understand that since this is a nonlinear problem, such an expression is not in general possible---I calculated the analytical toy example when $f(x)=e^{-ax^2}$, in which case $\mathcal{F}[f](q\,;n,k) = \{\textrm{an ugly polynomial in }q\}\times\left(\hat{F}(q)\right)^{1/k}$.
However, if I am able to make certain assumptions about my function $f$ then I might be able to make some approximations that could lead to a useful approximate/asymptotic analytical relationship. I'm at a loss for how to go about this. I am approaching this from an applied math/physics standpoint, so certain niceness assumptions about $f$ can be taken as given, like smoothness, square-integrability of all derivatives, etc.
Are there certain relevant theorems or conditions I should be looking for my function to satisfy (e.g. peakedness, periodicity---would this be cleaner for Fourier series?), or techniques I could try to get an approximate expression? I tried looking in a few of my texbooks and on the internet, and I didn't find much. If there is a large well-established literature on this suitable for a non-math grad student, you could also refer that to me in the comments. Thanks!