Spectral decomposition of locally stationary random processes

The notion of a locally stationary process is introduced by Silverman in [ l j . This is a new kind of a random process generalizing the notion of a weakly station­ ary process. Let {x(t)}, teR1bsa random process, generally complex, with vanishing mean value and finite covariance function R(s, t) = E{x(s) x(r)} on Wj x Mu where x(t) is the complex conjugate to x(r). The author of [ l j says that the random process {x(t)}, telRi, is locally stationary in the wide sense, or has a locally stationary covariance, if its covariance can be written as