Harmonic analysis meets stationarity: A general framework for series expansions of special Gaussian processes

In this paper, we present a new approach to derive series expansions for some Gaussian processes based on harmonic analysis of their covariance function. In particular, a new simple rate-optimal series expansion is derived for fractional Brownian motion. The convergence of the latter series holds in mean square and uniformly almost surely, with a rate-optimal decay of the rest of the series. We also develop a general framework of convergent series expansion for certain classes of Gaussian processes. Finally, an application to functional quantization is described