On the convergence rate at infinity of the Fourier transform of the standard bump function

Question

My question is concerned with the Fourier transform of the standard bump function, $\phi(x)=e^{-\frac{1}{x^2-1}}$ if $x\in (-1,1)$ and equal to $0$ if otherwise. As known, as $\phi$ has compactly supported in $[-1,1]$, the fourier transform ${\phi ^{\operatorname{ft} }}\left( t \right)$ tends to zero with rapidly rate at infinity. In Wikipedia at link http://en.wikipedia.org/wiki/Bump_function, there is a result that $|{\phi ^{\operatorname{ft} }}\left( t \right)| = O({\left| t \right|^{ - \frac{3}{4}}}{e^{ - \sqrt {\left| t \right|} }})$ as $\left|t\right|\to +\infty$. Actually, I want to find a complete proof for this result. Can someone give me a paper which was published has concerned this result, or a hint for proof of this result? Thank you for helping.

Answer 1

I found this article of Steven Johnson where it is perfectly explained how to apply the saddle point method in order to get the wanted inequality: $$\hat{\phi}(t)\ll |t|^{-3/4} e^{-\sqrt{|t|}}.$$