# 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 $f \in C^{1}(\mathbb{R})$

• This is probably false: there must exist compactly supported $C^1$-functions whose Fourier transform is not integrable. I'll look for an example. Apr 21 '11 at 6:13
• Does page 4 of the following link help? math.unc.edu/Faculty/met/s14.pdf 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 $\mathbb{R}^{n}$ Apr 21 '11 at 9:07
• @Suvrit: The link is broke. Could you please write out the author and title of the paper?
– Hans
Feb 19 '19 at 0:59