Suppose we have a function $f(x)$ such that $$|\frac{d^n}{dx^n}f(x)| \leq C(1 + |x|)^{-n}$$ Take the Fourier transform $\hat{f}(\xi)$ and consider the function $g(\xi) = \chi(\xi)\hat{f}(\xi)$, where $\chi(\xi)$ is a smooth function supported away from the origin and $\chi(\xi) \equiv 1$ outside $[-1, 1]$. What can we say about the the decay properties of the inverse Fourier transform of $g(\xi)$?
Edit: This is not a homework assignment. I am currently attending a course on Fourier analysis, and we used a cut-off function in a calculation today. I am trying to play around a bit with this concept. Thanks for your help.