Asymptotic eigenfunctions and eigenvalues of a homogeneous integral equation

The eigenfunctions and eigenvalues of a certain integral equation are of importance in the Karhunen-Loéve expansion of second-order stationary random functions. In this note the asymptotic eigenfunctions and eigenvalues of this integral equation are derived for the case where the kernel is the Fourier transform of a rational function of
\omega^2
.