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 $)