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Consider the inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$. My question is, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I understand that this is not exactly a research-level question, but the question appears a bit hard. Any help is appreciated.

Edit: Initially I had asked about the inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$, which was answered by the comments below. I looked up the formulae for complex powers of Euclidean norm of $(\xi_1, \xi_2)$, but the methodology there does not seem to apply to the edited question.

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  • $\begingroup$ Take Fourier transform in the sense of distributions; see standard textbooks on PDE or distributions. $\endgroup$
    – Ben McKay
    Commented Jun 9, 2015 at 15:48
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    $\begingroup$ entry 311 in the table at en.wikipedia.org/wiki/Fourier_transform $\endgroup$
    – Carlo Beenakker
    Commented Jun 9, 2015 at 15:53
  • $\begingroup$ There's a proof on p. 21 of Wolff's notes: math.ubc.ca/~ilaba/wolff/notes_march2002.pdf. The second function you ask about isn't as nice, but there is an explicit formula for the Fourier transforms of (complex) powers of the Euclidean norm of $(\xi_1,\ldots,\xi_d)$. $\endgroup$
    – Brendan Murphy
    Commented Jun 9, 2015 at 15:58

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