Hilbert transform - Knowlpedia

Definition

For $f$ in the Schwartz space on $\mathbb R$ , its Hilbert transform is

Hf(x)=\frac1\pi\operatorname{p.v.}\!\int_{\mathbb R} \frac{f(x-t)}{t}\,dt.

The operator extends uniquely to a bounded linear map on $L^2(\mathbb R)$ .

Fourier multiplier

With the Fourier convention $e^{-2\pi i x\xi}$ ,

\widehat{Hf}(\xi)=-i\,\operatorname{sgn}(\xi)\widehat f(\xi).

This identity proves $L^2$ -boundedness using the Plancherel theorem.

Harmonic conjugates

If $u$ is the Poisson extension of boundary data $f$ , then $Hf$ , up to the chosen sign, is the boundary value of a harmonic conjugate. Consequently the Dirichlet-to-Neumann operator is $f\mapsto H[-f']$ .

References

Loukas Grafakos, Classical Fourier Analysis, 3rd ed., Springer, 2014. DOI record.