  This paper studies Galerkin approximations applied to the Zakai equation of
stochastic filtering. The basic idea of this approach is to project the
infinite-dimensional Zakai equation onto some finite-dimensional subspace
generated by smooth basis functions; this leads to a finite-dimensional system
of stochastic differential equations that can be solved numerically. The
contribution of the paper is twofold. On the theoretical side, existing
convergence results are extended to filtering models with observations of
point-process or mixed type. On the applied side, various issues related to the
numerical implementation of the method are considered; in particular, we
propose to work with a subspace that is constructed from a basis of Hermite
polynomials. The paper closes with a numerical case study.
