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

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 $\Vert f^{+}-g^{+}{\Vert }_{H_2}$ is controlled by $\Vert f-g{\Vert }_{L_1}$ and $\Vert \log f-\log g{\Vert }_{L_1}$.

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• Quantitative results on continuity of the spectral factorization mapping