Resolvent Kernels of Green’s Function Kernels and Other Finite-Rank Modifications of Fredholm and Volterra Kernels

Many important Fredholm integral equations have separable kernels which are finite-rank modifications of Volterra kernels. This class includes Green’s functions for Sturm—Liouville and other two-point boundary-value problems for linear ordinary differential operators. It is shown how to construct the Fredholm determinant, resolvent kernel, and eigenfunctions of kernels of this class by solving related Volterra integral equations and finite, linear algebraic systems. Applications to boundary-value problems are discussed, and explicit formulas are given for a simple example. Analytic and numerical approximation procedures for more general problems are indicated.