Szegő Mapping and Hermite--Padé Polynomials for Multiple Orthogonality on the Unit Circle
Rostyslav Kozhan, Marcus Vaktnäs
TL;DR
The paper develops a unified framework for Laurent multiple orthogonal polynomials on the unit circle, showing that these objects are solutions to a general two-point Hermite–Padé problem for Carathéodory-type functionals. It derives Szegő-type nearest-neighbor recurrences, compatibility relations among recurrence data, Heine determinantal representations, and Christoffel–Darboux formulas, all in the Laurent setting with dual Type II and Type I problems. A central contribution is the Szegő mapping and Geronimus relations that connect circle-based Laurent MOPs to real-line MOPs, clarifying the unit-circle versus real-line orthogonality correspondence. These results provide explicit determinant and recurrence frameworks, enabling analysis of normality, zeros, and spectral properties in a broad multi-measure context. The work thus unifies and extends classical OPUC/MOPUC theory with real-line MOP via a two-point Hermite–Padé lens, with potential applications in approximation and spectral analysis on the unit circle.
Abstract
We investigate generalized Laurent multiple orthogonal polynomials on the unit circle satisfying simultaneous orthogonality conditions with respect to $r$ probability measures or linear functionals on the unit circle. We show that these polynomials can be characterized as solutions of a general two-point Hermite--Padé approximation problem. We derive Szegő-type recurrence relations, establish compatibility conditions for the associated recurrence coefficients, and obtain Christoffel--Darboux formulas as well as Heine-type determinantal representations. Furthermore, by extending the Szegő mapping and the Geronimus relations, we relate these Laurent multiple orthogonal polynomials to multiple orthogonal polynomials on the real line, thereby making explicit the connection between multiple orthogonality on the unit circle and on the real line.