## Hilbert transform

### Partial Function

 definiendum $H: (\mathbb C\to\mathbb C)\to(\mathbb C\to\mathbb C)$ definiendum $H(f):=y\mapsto \frac{1}{\pi}\cdot\mathcal P\int_{-\infty}^\infty\frac{f(x)}{y-x}\,\mathrm dx$

### Discussion

$(H(f))=-f$

The Hilbert transform commutes with the Fourier transform up to a simple factor and is an anti-self adjoint operator relative to the duality pairing between $L^p(\mathbb R)$ and the dual space $L^q(\mathbb R)$.

It is also used in the Kramers–Kronig relation/Sokhotski–Plemelj theorem to express the imaginary part of an analytic function in terms of its real part (or the other way around). This works because they get “mixed up” in the Cauchy integral formula which introduces a factor of $\frac{1}{i}$. The principal value is taken to push the complex line integral on the real line.

#### Reference

Wikipedia: Hilbert transform