Wiener–Hopf factorization

Remarks

The functions $1 - \hat k_\pm$ extend holomorphically to the corresponding half-planes, because $k_\pm$ vanish on a half-line. When $\nu = 0$, a continuous branch of $\log\bigl(1 - \hat k\bigr)$ on $\R$ tends to $0$ at infinity; splitting it into an upper and a lower part and exponentiating gives the two factors. The difficult point is that the split pieces again come from $L^1$ functions, which uses Wiener's lemma on absolutely convergent Fourier integrals. For $\nu \ne 0$ one first divides by $r^\nu$, whose winding number is $\nu$.

Example: $k(x) = 2 e^{-x}$ for $x > 0$ and $k = 0$ otherwise gives $1 - \hat k(\xi) = 1 - \frac{2}{1 - i\xi} = \frac{\xi - i}{\xi + i} = r(\xi)$. So $\nu = 1$, $k_\pm = 0$, and $W$ is injective with range of codimension $1$.

The hypothesis $1 - \hat k \ne 0$ on $\R$ is needed: for $k(x) = \frac{1}{2} e^{-\abs{x}}$ the symbol $\frac{\xi^2}{1 + \xi^2}$ vanishes at $0$, and then $W$ is not Fredholm on $L^p(0, \infty)$.

Going deeper

For a student

The theorem is the continuous analogue of the theory of Toeplitz operators, where the symbol lives on the unit circle instead of the real line. It turns the Wiener–Hopf method into a complete solvability theory for kernels in $L^1(\R)$. The sign convention of the Fourier transform matters. With the kernel $e^{-i\xi x}$ the half-planes swap roles and the winding number changes sign, so the index becomes $+\nu$. For systems of equations the symbol is a matrix function, and a single winding number is replaced by several partial indices, which is the matrix Wiener–Hopf factorization.