Here's a link to the paper: https://arxiv.org/abs/1610.03031
Ignoring the quite unconventional formatting (single column, figures at the end with the references), I'm not entirely convinced by the plots at the end of the paper. Specifically, their examples of "odd" Fourier transforms look quite normal. Specifically, the noise level (how it looks like a power function) is related to the "granulation background" of the star and is predicted by asteroseismic stellar models (the equations are ugly but they do exist). Normally you remove them from your transforms before you do anything with them. To be fair, they did say they "subtracted a smooth spectrum" but the noise level doesn't look flat enough to me. Also, their Fourier transforms are in super funky units -- in asteroseismology you generally want to use PSFs (Power Spectral Frequency, where the y axis is P/Hz).