Causality, realizability, and separability in reproducing kernel Hillbert space

Here we describe results for obtaining finite dimensional linear state space realizations for solutions to a generalized innovations representation problem: to characterize finite dimensional realizability of a causally invertible transformation which relates two given random processes. The solution is determined on a reproducing kernel Hilbert Space, (RKHS). where the reproducing kernel is the covariance function of one of the given random processes, u(??). This investigation succeeds in characterizing finite dimensional linear RKHS (innovations) realizations for the other random process, y(??), in terms of u(??). with a separability property of both the reproducing kernel and the covariance of the given random process, y(??).