Consider a bump function $f\in C_0^\infty(\mathbb{R})$ with the property \begin{align} f(x) = \begin{cases} 1\qquad \forall x\in (a,b)\,, \\ 0\qquad \forall x\in \mathbb{R}\setminus(a-\epsilon,b+\epsilon) \end{cases} \end{align} for some $\epsilon>0$. Basically this is just a constant function on an interval $(a,b)$ and interpolates to zero smoothly and monotonically within some small window $\epsilon$ such that $f$ is smooth and compactly supported. There are two "scales" available in the function: $\Delta = b-a>0$ and $\epsilon>0$.
Question: how much is known about the Fourier transform in terms of $\Delta,\epsilon$ --- in particular, the asymptotic behaviour --- in terms of these two numbers?
From Wikipedia (which I guess is based on stationary phase approximation), the case of the standard bump function using $e^{-\frac{1}{1-x^2}}$ is known to have asymptotics $O(|k|^{-3/4}e^{-\sqrt{k}})$. So it decays faster than any polynomial, but it does have "non-analytic" part due to the $1/|k|$. What I am trying to understand is the expected behaviour of the function $f$ in terms of $\Delta,\epsilon$.
On the one hand, Gaussian functions $g(x) = e^{-x^2/\sigma^2}$ (which has only one scale determined by its width $\sigma$) is smooth and not compactly supported, and the Fourier transform is also a Gaussian $\tilde{g}(k)\sim e^{-\sigma^2 k^2}$ with width $\sigma^{-1}$. On the other hand, "distributions" like the rectangular function $r(x)$ is compactly supported but discontinous at the edge; it has sinc function as its Fourier transform, so $\tilde{r}(k)\sim \sin(k \Delta )/k$, which gives $O(1/|k|)$. My intuition (from doing physics) is that if $f\in C^\infty_0(\mathbb{R})$ as required above, then it should be "in between": \begin{align} \tilde{f}(k) \sim O(|k|^{-\alpha}e^{-k^{\beta}}) \end{align} where, if I have to guess, $0<\alpha <1$ and $0< \beta <1$. Therefore, I can ask a more precise question: Is this guess correct and/or well-known?
If it is well-known, it would be great if you could point to a paper/book/etc. If it is not correct and there is an argument or counterexample, that would be great. If it is somehow trivial (and I am not seeing it) I would appreciate any hints on the strategy. Ideally it would be nice to know some kind of dependence of $\alpha,\beta$ with $\Delta,\epsilon$, but I think it may not be the case and the latter just appear in the Fourier coefficients.