Quantitative results on continuity of the spectral factorization mapping

TL;DR

This paper investigates the stability of spectral factorization, specifically how the difference between spectral factors relates to the differences in original functions and their determinants in various norms.

Contribution

It provides quantitative bounds on the continuity of the spectral factorization mapping in terms of $L_1$ and $H_2$ norms, advancing understanding of its stability properties.

Findings

Established bounds linking $ orm{F^+ - G^+}_{H_2}$ to $ orm{F-G}_{L_1}$ and $ orm{ race( abla F)}_{L_1}$

Quantified the stability of spectral factorization under perturbations of the original matrix functions

Extended previous qualitative results to explicit quantitative estimates.

Abstract

The spectral factorization mapping $F\to F^+$ puts a positive definite integrable matrix function $F$ having an integrable logarithm of the determinant in correspondence with an outer analytic matrix function $F^+$ such that $F = F^+(F^+)^*$ almost everywhere. The main question addressed here is to what extent $\|F^+ - G^+\|_{H_2}$ is controlled by $\|F-G\|_{L_1}$ and $\|\log \det F - \log\det G\|_{L_1}$.