  The inclusion of spatial smoothing in finite-dimensional particle-based
Hamiltonian reductions of the Vlasov equation are considered. In the context of
the Vlasov-Poisson equation (and other mean-field Lie-Poisson systems),
smoothing amounts to a convolutive regularization of the Hamiltonian. This
regularization may be interpreted as a change of the inner product structure
used to identify the dual space in the Lie-Poisson Hamiltonian formulation. In
particular, the shape function used for spatial smoothing may be identified as
the kernel function of a reproducing kernel Hilbert space whose inner product
is used to define the Lie-Poisson Hamiltonian structure. It is likewise
possible to introduce smoothing in the Vlasov-Maxwell system, but in this case
the Poisson bracket must be modified rather than the Hamiltonian. The smoothing
applied to the Vlasov-Maxwell system is incorporated by inserting smoothing in
the map from canonical to kinematic coordinates. In the filtered system, the
Lorentz force law and the current, the two terms coupling the Vlasov equation
with Maxwell's equations, are spatially smoothed.
