Rigorous theory for AAA-least squares (AAALS) on general domains

Develop a rigorous convergence and accuracy theory for AAA-least squares (AAALS), extending beyond special cases (disk or half-plane), and delineate conditions under which the method succeeds or fails.

Background

AAALS leverages AAA-selected exterior poles and least-squares fitting of real parts to solve Laplace problems efficiently. The authors explicitly state that no theory has been developed beyond limited settings and note counterexamples showing occasional failure. A comprehensive theory would clarify performance across geometries (e.g., corners, inlets) and data.