Utilization and improvement of AAA-derived information outside reflection domains

Ascertain whether information contained in AAA rational approximations r(z) of the Schwarz function S(z), computed from samples on an analytic arc Γ with S(z)=\bar z on Γ, can be usefully utilized outside numerically identified reflection domains, and determine whether alternative computational methods can improve upon this information.

Background

The paper computes the Schwarz function S(z) numerically using the AAA rational approximation algorithm applied to samples on an analytic curve Γ where S(z)=\bar z on Γ. Reflection domains are identified by testing the involution property \overline{S(\overline{S(z)})}=z and visualized via contours of |\overline{r(\overline{r(z)})}-z|. The authors observe that AAA approximations r(z) provide reliable reflections within green regions, and often contain accurate information in white regions as r(z) may approximate other branches of S(z).

However, it is unclear whether and how this information outside reflection domains can be exploited or enhanced. The authors highlight that numerical methods for tracking multivalued analytic functions exist but are not very advanced, motivating investigation into methodologies that can utilize or improve the branch information encoded by r(z).