Hilbert transform

Summary

The Hilbert transform is the linear operator on functions of a real variable given by convolution with $\frac{1}{\pi x}$ in the principal value sense. Let $u + iv$ be holomorphic on the upper half-plane and decay at infinity. Then the transform sends the boundary values of $u$ to those of $v$, so it is the operator of harmonic conjugation on the line. On the Fourier side it multiplies by $-i \sgn(\xi)$. It is the model example of a singular integral operator, and it appears in signal processing and in the Kramers–Kronig relations.