Geometric aspects of the Carathéodory extension problem

Both Euclidean geometric and non-Euclidean geometric descriptions for the regions of the solution set to the Carathéodory extension problem are discussed. In particular, it is shown that all the solution functions are located inside a Euclidean disk (the Weyl disk) and a hyperbolic disk (the Apollonian disk). The radius of each disk depends on the norm of the parametrizing Schur functions. Moreover, the so-called maximum entropy solution is exactly the hyperbolic center of the Apollonian disk, while the associated power spectral density function is the geometric mean center of the real region where the Weyl disk is projected to. Sensitivity analysis on the Weyl radius and Apollonian center due to the perturbation of the given finite moments is presented. The applications of these results to the spectrum estimation and inverse scattering problems are discussed.