Spectral relationship between a function and the reciprocal weight

Characterize the relationship between the spectra \(\sigma(f)\) and \(\sigma(1/f)\) for a continuous, nowhere-vanishing function \(f\) with absolutely summable Fourier coefficients.

Background

The paper observes that if $f$ is continuous and nowhere zero, Wiener’s lemma guarantees that $1/f$ has absolutely summable Fourier coefficients whenever $f$ does. Nevertheless, the distribution of the Fourier spectrum of the reciprocal is generally not understood.

The authors note that the spectra $\sigma(f)$ and $-\sigma(1/f)=\sigma(1/\overline f)$ must be highly overlapping in the sense of their woven-completeness characterization, and they rule out disjointness under the stated hypotheses. They leave open the more precise question of how the two spectra are related.