The Divergence of all Sampling-based Methods for Calculating the Spectral Factorization

Abstract

This paper investigates the possibility of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of the spectral densities. It will be shown that there exists no sampling-based method which depends continuously on the samples and which is able to approximate the spectral factor arbitrarily well for all continuous densities of finite energy. Instead, to any sampling-based approximation method there exists a large set of spectral densities so that the approximation method does not converge to the spectral factor for every spectral density in this set as the number of available sampling points is increased. Finally, the paper discusses shortly some consequences of these results. Namely, it mentions implications on the inner-outer factorization, it discusses algorithms which are based on a rational approximation of the spectral density, and it considers the Turing computability of the spectral factor.