Spectral Factorization of a Class of Matrix‐Valued Spectral Densities

Abstract

Recently, a necessary and sufficient uniform log-integrability condition has been established for the canonical spectral factorization mapping to be sequentially continuous. However, this condition, along with several other equivalent conditions, is not straightforward to verify. In this paper, we first derive a new set of easily checkable sufficient conditions which guarantee uniform log-integrability. Based on the newly derived conditions, we establish the existence of certain convergent rational approximations for a class of matrix-valued spectral densities. We then propose a new spectral factorization algorithm and provide convergence results. Our approach does not require the spectral density to be coercive. Numerical examples are given to illustrate the effectiveness and convergence of the proposed algorithm. In particular, we compute approximate spectral factors of the noncoercive and nonrational Kolmogorov and von Karman power spectra which arise in the study of turbulence.