Angelesco and AT systems on the Unit Circle

Abstract

We introduce the concept of Laurent multiple orthogonality on the unit circle and define Angelesco and AT systems in this setting. Using a generalized Andreief identity, we establish normality of all multi-indices for any such system, thereby ensuring existence and uniqueness of Laurent multiple orthogonal polynomials of type I and type II at every location. As an application, we demonstrate existence and uniqueness of the approximants for two natural two-point Hermite-Pad\'e problems -- type I and type II -- arising in the simultaneous rational approximation of $r$ Carath\'{e}odory functions.