Approximation and Convergence Behavior of Spectral Factorization Methods

Common methods for the calculation of the spectral factorization rely on an approximation of the given spectral density by a trigonometric polynomial and a subsequent spectral factorization of this polynomial. Since the approximative polynomial should be factorized, the approximation method must be positive. The first part of this paper studies such approximation methods and deduces limitation on the approximation rate for linear methods which arise from the required positivity. The second part states a lower and an upper bound on the error in the spectral factor induced by the approximation of the spectral density. They show the dependency of the error on the regularity of the stochastic process and on the approximative degree.