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I am confused about the 1d Fourier transform of the absolute value of $x$. In particular I'm confused why it doesn't show up on most Fourier-transform tables that I look for.

I'm using the convention $\mathcal{F}(f) = \int f(x) e^{-ikx} dx$. We can write $\| x \| = x \, \text{sgn}(x)$ and use the fact that $\mathcal{F}(\text{sgn}(x)) = \frac{2}{ik}$. Then:

$$ \begin{aligned} \mathcal{F}(\| x \|) &= \mathcal{F}(x \, \text{sgn}(x)) \\ &= (i \frac{d}{dk}) (\frac{2}{ik}) \\ &= -\frac{2}{k^2} \end{aligned} $$

But for whatever reason this Fourier transform doesn't seem to be on most of the tables I've looked up. For example, Wikipedia only lists $\| x \|^{\alpha}$ for $-1 < \alpha < 0$, and the same is true of the sources (Erdélyi, Kammler) that it cites. Plus I haven't found anyone asking the question on StackExchange! What's going on? Does nobody ever transform this function?

I think I have the right answer, but want to be sure that I'm not missing a delta-function term or something as I'm usually just guessing when it comes to those. So it would be nice to have a source that includes $\| x\|$ and, ideally, $\| x \|^{\alpha > 1}$ as well.

Alex Kritchevsky
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  • First, warning, this is a Fourier transform in the sense of distributions, do you know what it means? In particular, the Fourier transform of the sign function is more precisely the principal value distribution of $2/(ik)$. Second, you did not look sufficiently well on the site, it has been asked a lot of times ... – LL 3.14 Oct 05 '23 at 21:25
  • Well! If searching the site on their own search and and searching the site on Google doesn't find it, it's as good as missing and ought to be asked again, I figure. They should improve their search engine. And yes, I do know that it's a principle value, although it has proven quite hard to find any cogent reference on it. know about distributions and the fact that they are meaningful in terms of their integration against test functions, but I haven't yet understood if there's a setting in which I will actually get the wrong answer if I don't include the principle value. – Alex Kritchevsky Oct 05 '23 at 21:56

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