Orthogonal Polynomials, a Szeg\H{o}--Verblunsky Theorem and Baxter's Theorem on the Quaternionic Sphere

TL;DR

This paper develops a theory of orthogonal polynomials on the quaternionic unit sphere, extending classical results like Szeg"H{o} recurrences, Verblunsky's theorem, and Baxter's theorem to this new setting.

Contribution

It introduces quaternionic orthogonal polynomials based on $q$-positive measures and extends key classical theorems using matrix-valued analogues and new formulations.

Findings

Extended Szeg"H{o} recurrences to quaternionic setting

Proved a quaternionic version of the Szeg"H{o}--Verblunsky theorem

Established Baxter's theorem on the quaternionic sphere

Abstract

We introduce a theory of orthogonal polynomials on the unit sphere of the quaternions based on the notion of a $q$-positive measure (which originated in a work of Alpay, Colombo, the second author and Sabadini). The results we extend to this setting include the Szeg\H{o} recurrences, the Zeros Theorem for orthogonal polynomials, the Szeg\H{o}--Verblunsky theorem, and Baxter's theorem; to obtain these results, we utilise the Verblunsky coefficients (or Schur parameters) of Alpay, Colombo and Sabadini and a number of established results in the matricial setting. Our approach also requires matrix-valued analogues of Schur's recurrences for the coefficients of a Schur function and of Verblunsky's formula for the moments of a measure, which appear to be new.