A Generalized Pell's equation for a class of multivariate orthogonal polynomials
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929299774242816 |
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| author | Lasserre, Jean-Bernard Xu, Yuan |
| author_facet | Lasserre, Jean-Bernard Xu, Yuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_10668 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Generalized Pell's equation for a class of multivariate orthogonal polynomials Lasserre, Jean-Bernard Xu, Yuan Optimization and Control Probability 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. |
| title | A Generalized Pell's equation for a class of multivariate orthogonal polynomials |
| topic | Optimization and Control Probability |
| url | https://arxiv.org/abs/2307.10668 |