Wiener–Hopf equation

Summary

A Wiener–Hopf equation is a convolution integral equation on the half-line. The kernel depends only on $x - t$, but the unknown lives on $[0, \infty)$ and the integral runs over $[0, \infty)$. The cut at $0$ breaks the translation symmetry, so the Fourier transform alone does not solve it. It is solved by the Wiener–Hopf method. This method factors the symbol of the equation into pieces that extend holomorphically to the two half-planes. Such equations model radiative transfer in a half-space, diffraction by a half-plane, and optimal causal filtering.