The Hilbert transform is formally defined as
\begin{equation*}
Hf (x) \coloneqq \text{P.V.} \frac 1{\pi x} * f (x)=\frac 1\pi \int_{-\infty}^{+\infty} \frac{f(y)}{x-y}dx.
\end{equation*} Some important facts are listed as follows.
- The Fourier transform of the kernel
\begin{equation*}
\Fi\br{\text{P.V.} \frac{1}{\pi x} }=-i \sgn .
\end{equation*} - We have identities for the Hilbert transform:
\begin{equation*}
\begin{split}
H(Hf)=& -f,\\
H(fg)=& fHg+gHf+H(Hg Hf ),\\
H(fHf)=& \frac12 \br{(Hf)^2-f^2 } .
\end{split}
\end{equation*}
We gives out some classic Hilbert transform:
- $H (\sin x)= -\cos x, H(\cos x)=\sin x.$
- $H \br{\frac{1}{1+x^2}}=\frac{x}{1+x^2}, H\br{\frac{x}{1+x^2} }=\frac{-1}{1+x^2}.$
This lecture is unfinished yet, I will add more contents later. And anyone who offer the detailed calculation of the second one will be appreciated.