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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.

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    Are you sure you want $\|\omega \| = \sum \omega_i^2$? Are do you want $\|\omega\|^2 = \sum \omega+i^2$?2011-07-02
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    Quite a bit of this is also discussed at a more accessible/practical level (compared to Gelfand and Shilov) in M. Taylor's _Partial Differential Equations_, volume 1, Section 3.8. (In my copy pages 239 - 245).2011-07-02

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