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[2307.10668] A Generalized Pell's equation for a class of multivariate orthogonal polynomials
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[v1] Thu, 20 Jul 2023 07:47:51 UTC (23 KB)
[v2] Tue, 2 Apr 2024 13:00:21 UTC (25 KB)
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Mathematics > Optimization and Control
arXiv:2307.10668 (math)
[Submitted on 20 Jul 2023 (v1), last revised 2 Apr 2024 (this version, v2)]
Title:A Generalized Pell's equation for a class of multivariate orthogonal polynomials
View a PDF of the paper titled A Generalized Pell's equation for a class of multivariate orthogonal polynomials, by Jean-Bernard Lasserre (LAAS-POP and 2 other authors
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Abstract:We extend the polynomial Pell's equation satisfied by univariate Chebyshev polynomials on [--1, 1] from one variable to several variables, using orthogonal polynomials on regular domains that include cubes, balls, and simplexes of arbitrary dimension. Moreover, we show that such an equation is strongly connected (i) to a certificate of positivity (from real algebraic geometry) on the domain, as well as (ii) to the Christoffel functions of the equilibrium measure on the domain. In addition, the solution to Pell's equation reflects an extremal property of orthonormal polynomials associated with an entropy-like criterion.
| Comments: | To appear in Transactions of the Amercian Mathematical Society |
| Subjects: | Optimization and Control (math.OC); Probability (math.PR) |
| Report number: | Rapport LAAS n{\textdegree} 23207 |
| Cite as: | arXiv:2307.10668 [math.OC] |
| (or arXiv:2307.10668v2 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2307.10668
arXiv-issued DOI via DataCite
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Submission history
From: Jean Bernard Lasserre [view email] [via CCSD proxy][v1] Thu, 20 Jul 2023 07:47:51 UTC (23 KB)
[v2] Tue, 2 Apr 2024 13:00:21 UTC (25 KB)
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View a PDF of the paper titled A Generalized Pell's equation for a class of multivariate orthogonal polynomials, by Jean-Bernard Lasserre (LAAS-POP and 2 other authors
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