  This paper is concerned with the analysis of the kernel-based algorithm for
gain function approximation in the feedback particle filter. The exact gain
function is the solution of a Poisson equation involving a probability-weighted
Laplacian. The kernel-based method -- introduced in our prior work -- allows
one to approximate this solution using {\em only} particles sampled from the
probability distribution. This paper describes new representations and
algorithms based on the kernel-based method. Theory surrounding the
approximation is improved and a novel formula for the gain function
approximation is derived. A procedure for carrying out error analysis of the
approximation is introduced. Certain asymptotic estimates for bias and variance
are derived for the general nonlinear non-Gaussian case. Comparison with the
constant gain function approximation is provided. The results are illustrated
with the aid of some numerical experiments.
