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Let $\alpha > 0$. I'm looking to better understand the Fourier transform of the function

$$f(x; \alpha) = \frac{1}{|x|\left(\mathrm{ln}|x|\right)^\alpha}$$

In particular, I would like to know whether it has an explicit representation and/or what kind of growth asymptotics it has. Since the Fourier transform of a product is the convolution of Fourier transforms, the question boils down to understanding what the Fourier transform of $\frac{1}{\left(\mathrm{ln}|x|\right)^\alpha}$ should be in general; the transform of $\frac{1}{|x|}$ is covered e.g. in here.

As of writing I'm still at square one of trying to find/work with the transform of $\frac{1}{\left(\mathrm{ln}|x|\right)^\alpha}$, so I'm really just asking for any and all help with this: feel free to give a full answer, but I'm also happy to hear about any good source which computes transforms of painful functions such as $\frac{1}{(\mathrm{ln}(|x|))^\alpha}$.

Cartesian Bear
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  • For what values of $\alpha$ do you believe that the FT exists? – Mark Viola Jan 18 '24 at 15:47
  • Just FYI ... The Fourier Transform of $\log(|x|)/|x|$ is $$\log^2(|k|)+2\gamma \log(|k|)+\gamma^2-\pi^2/12$$ – Mark Viola Jan 18 '24 at 15:53
  • @MarkViola I would hope that for all $\alpha > 0$, but right now I have nothing to base this on. I might be able to compute some numerics later on, but that will take a day or two. – Cartesian Bear Jan 18 '24 at 16:00
  • Are you defining the Fourier transform as a tempered distribution? If not, then $e^{ikx}/(\log(|x|)^\alpha$ has a non-integrable singularity for $\alpha\ge 1$ at $|x|=1$ – Mark Viola Jan 18 '24 at 16:14
  • @MarkViola In this case, it is defined as a tempered distribution. – Cartesian Bear Jan 19 '24 at 09:17

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