Representation of strongly harmonizable periodically correlated processes and their covariances

This paper addresses the representation of continuous-time strongly harmonizable periodically correlated processes and their covariance functions. We show that the support of the 2-dimensional spectral measure is constrained to a set of equally spaced lines parallel to the diagonal. Our main result is that any harmonizable periodically correlated process may be represented in quadratic mean as a Fourier series whose coefficients are a family of unique jointly wide sense stationary processes; the corresponding family of cross spectral distribution functions may be simply identified from the two-dimensional spectral measure resulting from the assumption of strong harmonizability.