Does talking about the inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$ even make sense? If it does, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I am learning about the Fourier transform and understanding examples like these would help me a lot.
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The inverse looks something like this – creative Jun 09 '15 at 08:52
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@Loophole Thanks for your answer. I am looking for a bit more theoretical answer, but it is good to know that it exists. For example, I would be happy to infer theoretically that the inverse Fourier transform is smooth when $t \neq 0$. – anonymous Jun 09 '15 at 09:05
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For clarification, do you really want $1/\sqrt{\xi}$ or instead $1/\sqrt{|\xi|}$? – Willie Wong Jun 10 '15 at 14:53
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Subject to the above clarification, this may be a duplicate of ($n$-dimensional) Inverse Fourier transform of $\frac{1}{| \mathbf{\omega} |^{2\alpha}}$ – Jun 10 '15 at 22:51