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.
Best Answer
I won't include all details since you don't seem to want that...
Up to minus signs and constants your problem is the same as trying to find the Fourier transform of $f(\omega) = \frac{1}{\|\mathbf{\omega}\|^{2\alpha}}$. This distribution is radial and homogeneous of degree $-2\alpha$. You can use the scaling properties of the Fourier transform to show that this means that its Fourier transform is radial and homogeneous of degree $-n + 2\alpha$. Then you can show that this only happens when this Fourier transform is of the form $c_\alpha \frac{1}{\|\mathbf{\xi}\|^{n - 2\alpha}}$.
So it remains to determine $c_\alpha$. For this you can use Plancherel's theorem in conjunction with the fact that $e^{-\pi |\omega|^2}$ is its own Fourier transform, so that you have $$\int_{R^n} \frac{1}{\|\mathbf{\omega}\|^{2\alpha}}e^{-\pi|\omega|^2}\,d\omega = c_{\alpha} \int_{R^n} \frac{1}{\|\mathbf{\xi}\|^{n - 2\alpha}}e^{-\pi|\xi|^2}\,d\xi$$ You can turn both integrals into one dimensional integrals using polar coordinates and then solve for $c_{\alpha}$. The result will be a ratio of gamma functions.