MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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? – 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

Here is a link to the paper about necessary conditions for the integrability of the Fourier transform:

It is stated in that paper that sufficient conditions for the integrability of the Fourier transform are given in the book

R. M. Trigub and E. S. Bellinsky, Fourier analysis and approximation of functions. Kluwer Academic Publishers, Dordrecht, 2004.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.