Orthogonal Polynomials, a Szegő--Verblunsky Theorem and Baxter's Theorem on the Quaternionic Sphere
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915993583878144 |
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| author | Gauntlett, Connor J. Kimsey, David P. |
| author_facet | Gauntlett, Connor J. Kimsey, David P. |
| contents | 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 Szegő recurrences, the Zeros Theorem for orthogonal polynomials, the Szegő--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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07787 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Orthogonal Polynomials, a Szegő--Verblunsky Theorem and Baxter's Theorem on the Quaternionic Sphere Gauntlett, Connor J. Kimsey, David P. Classical Analysis and ODEs Complex Variables Functional Analysis 42C05, 30G35, 47A57 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 Szegő recurrences, the Zeros Theorem for orthogonal polynomials, the Szegő--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. |
| title | Orthogonal Polynomials, a Szegő--Verblunsky Theorem and Baxter's Theorem on the Quaternionic Sphere |
| topic | Classical Analysis and ODEs Complex Variables Functional Analysis 42C05, 30G35, 47A57 |
| url | https://arxiv.org/abs/2605.07787 |