Spectral representations
Georg Lindgren, Holger Rootzen, Maria Sandsten · 2013
It may be fair to claim that the spectrum is the main instrument in the theory and application of stationary stochastic processes. The spectral decomposition of the covariance function gives a unique characterization of its properties in terms of a Fourier transform. An analogue is the Fourier transform of data, which predates the covariance approach. The view taken in this book is that data are observations of a stationary process. Both transforms represent a decomposition of a function (covariance function or data series, respectively) into a sum of cosine functions. In this chapter, we deal mainly with the Fourier transform of the covariance function, called the spectral density, but we also illustrate the corresponding transform of data. A thorough discussion of Fourier analysis of data will be given in Chapter 9.