# Smoothness and decay property of Fourier transformation

If my memory serves I have heard something like "the less smooth your function $f$ is, the worse its Fourier transform $\hat{f}$ decay because its Fourier transform $\hat{f}$ needs more waves of high frequency". I now would like to formulate the claim above properly.

I am interested in relation of smoothness and decay property of Fourier transformation. I think this can be rigorously shown by the fundamental properties of Fourier transformation:

$\widehat{\partial_{x}^n f}(\xi)=(i\xi)^n\hat{f}(\xi)$ and $\widehat{(i\xi)^n f}(\xi)=\partial_{x}^n\hat{f}(\xi)$

From these equalities how can one conclude smoothness and decay property of Fourier transformation?

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If $f$ has an integrable (weak) derivative of order $n$, then the map $\xi\mapsto \xi^n\widehat f(\xi)$ is in $C_0$, the space of continuous functions which go to $0$ at infinity. This is because we expressed $\xi^n\widehat f(\xi)$ as a Fourier transform of an integrable function.
We deduce that $\widehat f$ decays at least like $|\xi|^{-n}$.