I know how the convolution of a distribution is defined. But when my teacher defined the Hilbert transform of a $f\in\mathcal{C}^1(\Bbb R)\cap L^1(\Bbb R)$ as $$ Hf(x):=\frac1{\pi}\operatorname{p.v.}\int_{\Bbb R}\frac1s f(x-s)\,ds $$ he said that $Hf$ is the convolution between $f$ and $\frac1{\pi}\operatorname{p.v.}\frac1x$, where $\operatorname{p.v.}\frac1x$,is the distribution which sends $$ \phi\mapsto\operatorname{p.v.}\int_{\Bbb R}\frac1x\phi(x)\,dx\;\;. $$ Did I misunderstood my teacher? Or maybe there exists a definition of the convolution between a function and a distribution?
Many thanks