  The paper deals with kernel density estimates of filtering densities in the
particle filter. The convergence of the estimates is investigated by means of
Fourier analysis. It is shown that the estimates converge to the theoretical
filtering densities in the mean integrated squared error under a certain
assumption on the Sobolev character of the filtering densities. A sufficient
condition is presented for the persistence of this Sobolev character over time.
Both results are extended to partial derivatives of the estimates and filtering
densities.
