Rigorous description of the approximation made by the ad hoc modification to the Kramers-Kronig constrained...

The Kramers-Kronig constrained variational analysis is a universal method developed for the spectral parameterization of virtually all types of linear-response optical quantities (reflectivity, transmission, complex refractive index, etc.). It relies essentially on the analytical expression resulting from the Hilbert transform of a basic triangular-shaped input function. A post-processing modification to the obtained analytical expression was introduced ad hoc in Mayerhöfer and Popp (2019), without mathematical grounds, to achieve better results and decrease the average errors by about three orders of magnitude. The modification was commented but not explained in Rousseau et al. (2021). The present communication follows on the mentioned chain of papers by providing a rigorous mathematical explanation to the effect of the ad hoc post-processing modification, with accompanying visual examples. The modification approximately corrects a function with inverted parity, and, at the same time, distorts the investigated triangular-shaped input function. In parallel, the present communication also highlights and exploits a parity-related invariance property of the Hilbert transform. To the best of the author's knowledge, this invariance property has not been reported elsewhere.