Fredholm resolvents, Wiener-Hopf equations, and Riccati differential equations

We shall show that the solution of Fredholm equations with symmetric kernels of a certain type can be reduced to the solution of a related Wiener-Hopf integral equation. A least-squares filtering problem is associated with this equation. When the kernel has a separable form, this related problem suggests that the solution can be obtained via a matrix Riccati differential equation, which may be a more convenient form for digital computer evaluation. The Fredholm determinant is also expressed in terms of the solution to the Riccati equation; this formula can also be used for the numerical determination of eigenvalues. The relations to similar work by Anderson and Moore and by Schumitzky are also discussed.