Approximate simultaneous orthogonal expansions. Applications to mean-square estimation and signal detection problems

Approximate simultaneous orthogonal expansions of two second-order stochastic processes are defined and their convergence is showed. The technique is based on the Rayleigh-Ritz method to solve the homogeneous equation involving both covariance kernels simultaneously. Finally, two specific applications of these finite expansions are given: in the Gaussian estimation and detection problems.