Angelesco and AT systems on the Unit Circle
Rostyslav Kozhan, Marcus Vaktnäs
TL;DR
The paper develops Laurent multiple orthogonality on the unit circle and proves that Angelesco and AT systems yield $\phi$-normality for every multi-index $\bm n$, guaranteeing existence and uniqueness of both type I and II Laurent MOPs at all locations. It establishes this via a generalized Andreief determinant identity and moment matrices, providing determinantal representations and linking the orthogonality to two-point Hermite–Padé problems for Carathéodory functions. The authors treat parity cases comprehensively (even, odd, and mixed), showing that the HP problems have unique solutions that correspond to the Laurent MOPs, and they derive explicit forms for the associated second-kind polynomials and half-power extensions. The results extend the MOP framework to the unit circle with robust normality results for Angelesco and AT systems, enabling canonical two-point rational approximants to Carathéodory data and paving the way for further spectral and approximation-theoretic applications.
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-Padé problems -- type I and type II -- arising in the simultaneous rational approximation of $r$ Carathéodory functions.