The solution of a homogeneous Wiener-Hopf integral equation occurring in the expansion of second-order stationary random functions

In many of the applications of probability theory to problems of estimation and detection of random functions an eigenvalue integral equation of the type \begin{equation} \phi(x) = \lambda \int_0^T K(x - y)\phi(y) dy, \qquad 0 \leq x \leq T, \end{equation} is encountered where
K(x)
represents the covariance function of a continuous stationary second-order process possessing an absolutely continuous spectral density. In this paper an explicit operational solution is given for the eigenvalnes and eigenfunctions in the special but practical case when the Fourier transform of
K(x)
is a rational function of
\omega^2
, i.e., \begin{equation} K(x) \doteqdot G(s^2) = \frac{N(s^2)}{D(s^2)}, \qquad s=i\omega, \end{euation} in which
N(s^2)
and
D(s^2)
are polynomials in
s^2
. It is easy to show by elementary methods that the solutions are of the form \begin{equation} \phi(x)= \sum C_r e^{-\alpha_r x} \cos (\beta_r x + \gamma_r), \end{equation} the constants
C_r, \alpha_r, \beta_r
, and
\gamma_r
being linked together by the integral equation. It is precisely the labor involved in their determination that in practice often causes the problem to assume awesome proportions. By means of the results given herein, this labor is diminished to the irreducible minimum-the solving of a transcendental equation.