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I am working with Strichartz's "A Guide to Distribution Theory and Fourier Transforms" (self-study -> not a homework question). He says none of the distributions that correspond to $1/|x|$ are non-negative. I would appreciate some help in order to understand why that is.

( we say the distribution f is non-negative if $\langle f,\psi \rangle \ge 0 $ for every test function $\psi$ that is non-negative with smooth test function that are compactly supported. Further we say a distribution $T$ corresponds to $1/|x|$ iff $T(\varphi)=∫\varphi(x)|x|dx$ for every test function $\varphi$ with $\varphi(0)=0 $)

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    What are the distributions corresponding to $1/|x|$? Does he correspond different distributions to the same function?2012-09-19
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    @HuiYu yes, he definitely means for different distribution to correspond to the same function ! unfortunately understanding the whole set of distributions that can be associated with $1/|x|$ is part of what I struggle with. So far he only introduced the correspondence via integration which works because of the compact support requirement. (i.e. $\langle f, \psi \rangle = \int_\Omega f(x) \psi(x) \;dx)$2012-09-19
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    @Beltrame: I have checked the book online: The author says a distribution $T$ corresponds to $1/|x|$ iff $T(\varphi)=\int\frac{\varphi(x)}{|x|} dx$ for every test function $\varphi$ **with $\varphi(0)=0$**. You should mention this in your question.2012-09-19
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    @Vobo thanks, ammended !2012-09-21

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