Wiener–Hopf method

Definition

Consider the Wiener–Hopf equation $\varphi(x) - \int_0^\infty k(x - t)\, \varphi(t)\, dt = f(x)$ for $x \ge 0$, with $k \in L^1(\R)$. Write $\hat g(\xi) = \int_{\R} g(x)\, e^{i \xi x}\, dx$ for the Fourier transform. For this convention, the transform of a function supported in $[0, \infty)$ is holomorphic in the upper half-plane $\operatorname{Im} \xi > 0$. The transform of a function supported in $(-\infty, 0]$ is holomorphic in the lower half-plane. The Wiener–Hopf method is the following procedure.

1. Extend $\varphi$ and $f$ by $0$ to $x < 0$, and let $\psi(x) = \int_0^\infty k(x - t) \varphi(t)\, dt$ for $x < 0$ and $\psi(x) = 0$ for $x \ge 0$. Then $\varphi - k * \varphi = f - \psi$ on all of $\R$.
2. Take Fourier transforms: $\bigl(1 - \hat k(\xi)\bigr)\, \hat\varphi(\xi) = \hat f(\xi) - \hat\psi(\xi)$, where $\hat\varphi$ and $\hat f$ are upper functions and $\hat\psi$ is a lower function.
3. Factor the symbol as $1 - \hat k = a_- \, a_+$. Here $a_+$ and $1/a_+$ are holomorphic and bounded in the upper half-plane. Also $a_-$ and $1/a_-$ are holomorphic and bounded in the lower half-plane.
4. Split $\hat f / a_- = g_+ + g_-$ into an upper and a lower part, for example with Cauchy integrals.
5. The equation becomes $a_+ \hat\varphi - g_+ = g_- - \hat\psi / a_-$. The left side extends to the upper half-plane and the right side to the lower half-plane, so together they define an entire function. Growth bounds and Liouville's theorem show that it is a polynomial, often $0$.
6. Then $\hat\varphi = (g_+ + P)/a_+$ for that polynomial $P$, and $\varphi$ is recovered by the inverse Fourier transform.