On Wiener filtering of certain locally stationary stochastic processes

We study linear minimum mean squared error filters for continuous-time second-order stochastic processes that are locally stationary in Silverman's sense. We show that the optimal filter is rarely locally stationary even when the covariance functions have Gaussian shape. Using Mehler's formula we derive series expansions of the filter kernel for locally stationary covariances that are determined by Gaussians.