Quantitative results on continuity of the spectral factorization mapping in the scalar case

TL;DR

This paper investigates the stability and continuity properties of the spectral factorization mapping in the scalar case, focusing on how perturbations in the original function and its logarithm affect the spectral factor.

Contribution

It provides quantitative results on the continuity of the spectral factorization mapping with respect to specific norms in the scalar case.

Findings

Established bounds on the difference between spectral factors in terms of $L_1$ norms.

Analyzed the dependence of spectral factor stability on perturbations of the original function.

Contributed to understanding the robustness of spectral factorization in signal processing.

Abstract

In the scalar case, the spectral factorization mapping $f\to f^+$ puts a nonnegative integrable function $f$ having an integrable logarithm in correspondence with an outer analytic function $f^+$ such that $f = |f^+|^2$ almost everywhere. The main question addressed here is to what extent $\|f^+ - g^+\|_{H_2}$ is controlled by $\|f-g\|_{L_1}$ and $\|\log f - \log g\|_{L_1}$.