MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

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

Note: I first posted question on math.stackexchange and I got one reply, which was a bit helpful (I'm still trying to understand it fully), but did not explore the two solution cases that I mentioned. Besides I would appreciate additional insight on this question, which I hope its not too trivial for mathoverflow. Thanks.

I'd like to find the $n$-dimensional inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$ i.e. $$ \int_{\mathbb{R}^n} \frac{1}{ \| \mathbf{\omega} \|^{2\alpha}} e^{2 \pi i \mathbf{\omega}\cdot \mathbf{x} } d \mathbf{\omega} $$ where $\mathbf{x} = ( x_0 , x_1 , \cdots , x_n )$ is a spatial parameter in $\mathbb{R}^n$, $\mathbf{\omega} = ( \omega_0 , \omega_1 , \cdots , \omega_n )$, and $$ \| \omega\| = \omega_0^2 + \omega_1^2 + \cdots + \omega_n^2 $$ All I've been able to come up with in the one-dimensional case is that the integral $$ \int_{-\infty}^{+\infty} \frac{1}{ \| \omega \|^{2\alpha}} e^{2 \pi i \omega x } d \mathbf{\omega} $$ diverges because the lower power terms $\omega^p$ terms, for which $p < 2\alpha$, in expansion of the exponential $$ e^{2 \pi i \omega x } = \sum_{p = 0}^{\infty} \frac{(2 \pi i \omega x)^p}{p!} $$ do not prevent $\frac{1}{\| \omega \|^{2\alpha}}$ from blowing up at the origin.

I know that one possible way of regularizing this integral is to include a test function and consider the limit of the resulting integral, but I don't quite know how to do so. I've tried reading Gelfand and Shilov's Gneralized Functions vol 1 and while I understand bits of it on the whole its a bit heavy for me.

Based on the papers that I've read I know that there are two cases (the latter of which appears to me more general) and two solutions in each.

  • Case 1: 2$\alpha$ is an odd/even integer
  • Case 2: 2$\alpha$ is integer or otherwise

I'd appreciate help, if possible, coming up with both solutions.

share|cite|improve this question
@Olumide: in this case you should bump your Math.SE question. In anycase, Zarrax already gave you the answer to your question. Why not just follow it up there? – Willie Wong Jul 2 '11 at 11:57
Will do. I just assumed I'd find more professional mathematicians, and thus new perspectives here on MO. – Olumide Jul 2 '11 at 12:33

Let's start with a simple change of notation. Let me consider first on $\mathbb R^n_x$ the function $f_{\beta}(x)=\Vert x\Vert^{\beta-n}$ for $0<\beta< n$, which is locally integrable and homogeneous with degree $\beta-n$. Its Fourier transform is also a radial distribution, i.e. such that $$ \forall A\in O(n),\quad \hat{f_\beta}=\hat{f_\beta}\circ A, $$ which is homogeneous with degree $-\beta+n-n=-\beta.$ A direct computation shows that $$ \hat{f_\beta}(\xi)=\Vert \xi\Vert^{-\beta} \frac{\pi^{-\beta/2}\Gamma(\beta/2)}{\pi^{-(n-\beta)/2}\Gamma((n-\beta)/2)}. $$ There are various extensions of this formula to other values of $\beta$, but one has to pay attention that for $\beta\le 0$, the ``function" $f_\beta$ is not locally integrable. It is then necessary to define an homogeneous distribution with degree $\beta-n$ which coincides with $\Vert x\Vert^{\beta-n}$ on $\mathbb R^n\backslash\{0\}$, which is possible uniquely if $\beta$ is not a nonpositive integer.


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.