  We examine an application of the kernel-based interpolation to numerical
solutions for Zakai equations in nonlinear filtering, and aim to prove its
rigorous convergence. To this end, we find the class of kernels and the
structure of collocation points explicitly under which the process of iterative
interpolation is stable. This result together with standard argument in error
estimation shows that the approximation error is bounded by the order of the
square root of the time step and the error that comes from a single step
interpolation. Our theorem is well consistent with the results of numerical
experiments.
