How does the Hilbert transform work?

Suppose we have $X(t) = \Re(Z(t))$ and we assume that $Z(t)$ is holomorphic on the upper half plane. It seems the most profound way to approach this is to examine the tempered distribution $\operatorname{p.v.} \frac{1}{\pi t}$. What is the representation of this operator in Fourier space, and why is it tempered?