The method for exploiting stochastic smoothing techniques to develop dynamical recursive algorithms for the deterministic problem of d interpolation (optimal curve fitting) is shown. A reproducing kernel Hilbert space approach is used to develop an explicit correspondence between spline interpolation and linear least-squares smoothing of a particular zero-mean random process. This random process is shown to be the output of a white-noise-driven dynamical system whose parameters and initial conditions are fixed by the functional form chosen for the spline. A recursive algorithm is then derived for this (nonstandard) smoothing problem, and thus also for the original spline interpolation problem.

The theory of reproducing kernel Hilbert spaces is used to obtain a simple but formal expression for the likelihood ratio (LR) for discriminating between two Gaussian processes with unequal covariances, and to develop a test by which the formal expression can be checked for validity. This LR formula can be evaluated by working separately with each covariance, thus reducing the calculations for the random signal case to those for the simpler known signal problem. In contrast, all previous LR formulas for the unequal covariance problem seem to require calculations involving both covariances simultaneously.

We first show how reproducing kernel Hilbert space (RKHS) norms can be determined for a large class of covariance functions by methods based on the solution of a Riccati differential equation or a Wiener-Hopf integral equation. Efficient numerical algorithms for such equations have been extensively studied, especially in the control literature. The innovations representations enter in that it is they that suggest the form of the RKHS norms. From the RKHS norms, we show how recursive solutions can be obtained for certain Fredholm equations of the first kind that are widely used in certain approaches to detection theory. Our approach specifies a unique solution: moreover, the algorithms used are well suited to the treatment of increasing observation intervals.