# least condition for the Fourier transform to be integrable

I want to prove that if $f \in C^{1}(\mathbb{R})$ is compactly supported then its Fourier transform is integrable. I was able to prove the result for $f \in C^{2}(\mathbb{R})$ and compactly supported. I used the fact that if $f \in C^{2}(\mathbb{R})$, then $\hat{f}$ is bounded by $\frac{c}{1+{|x|}^{2}}$. So it is integrable. I failed to prove it if $f \in C^{1}(\mathbb{R})$

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This is probably false: there must exist compactly supported $C^1$-functions whose Fourier transform is not integrable. I'll look for an example. –  Denis Serre Apr 21 '11 at 6:13
Does page 4 of the following link help? math.unc.edu/Faculty/met/s14.pdf –  Suvrit Apr 21 '11 at 7:35
No, but page 6 does :), thanks a lot,... I'm trying to see now if this result can be upgraded to $\mathbb{R}^{n}$ –  jessica Apr 21 '11 at 9:07