The Hilbert transform and Hermite functions: a real variable proof of the \(L^2\)-isometry (Q944353)

The Hilbert transform of a function \(f\) defined on the real line is given by \[ Hf(x)=p.v.\frac1{\pi}\int_R\frac{f(x-t)}t \,dt:=\lim_{\varepsilon\rightarrow 0} \frac1{\pi}\int_{| t| >\varepsilon}\frac{f(x-t)}t \,dt. \] The author provides a proof of the fact that the Hilbert transform can be extended as an isometry to \(L^2\) and obtains this by real variable methods using the Hermite functions.