Math 6420 • PDE

The (generalized) Hilbert transform \(H\) of a periodic function
\(f\) can be defined as follows: For any simply connected two-dimensional domain \(\Omega\),
let \(\phi\) satisfy \(\nabla^2\phi=0\) with \(\nabla\phi\cdot\mathbf{n}=f\) on \(\partial\Omega\).
Then \(Hf=\nabla\phi\cdot\mathbf{t}\) on \(\partial\Omega\). (\(\mathbf{n}\) is the exterior unit
normal, \(\mathbf{t}\) is the unit tangent vector.)
\begin{enumerate}
\item[a)] Prove that \(H(Hf)=-f\).
\item[b)] Find a relationship between the Fourier transform and the Hilbert transform,
using a unit circle as the domain \(\Omega\).
\item[c)] What is the Hilbert transform of \(f(x)=a\sin(x)+b\cos(2x)\)?
\end{enumerate}