  We derive a reduced-order state estimator for discrete-time infinite
dimensional linear systems with finite dimensional Gaussian input and output
noise. This state estimator is the optimal one-step estimate that takes values
in a fixed finite dimensional subspace of the system's state space ---
consider, for example, a Finite Element space. We then derive a Riccati
difference equation for the error covariance and use sensitivity analysis to
obtain a bound for the error of the state estimate due to the state space
discretization.
