Two-dimensional diffusion orthogonal polynomials ordered by a weighted degree
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912139659182080 |
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| author | Orevkov, Stepan |
| author_facet | Orevkov, Stepan |
| contents | We study the following problem: describe the triplets $(Ω,g,μ)$, $μ=ρ\,dx$, where $g= (g^{ij}(x))$ is the (co)metric associated with the symmetric second order differential operator $L (f) = \frac{1}ρ\sum_{ij} \partial_i (g^{ij} ρ\partial_j f)$ defined on a domain $Ω$ of $\mathbb R^d$ and such that there exists an orthonormal basis of $\mathcal L^2(μ)$ made of polynomials which are eigenvectors of $L$, where the polynomials are ranked according to some weighted degree.
In a joint paper with D. Bakry and M. Zani this problem was solved in dimension 2 for the usual degree. In the present paper we solve it still in dimension 2, but for a weighted degree with arbitrary positive weights. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_04949 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Two-dimensional diffusion orthogonal polynomials ordered by a weighted degree Orevkov, Stepan Algebraic Geometry Classical Analysis and ODEs We study the following problem: describe the triplets $(Ω,g,μ)$, $μ=ρ\,dx$, where $g= (g^{ij}(x))$ is the (co)metric associated with the symmetric second order differential operator $L (f) = \frac{1}ρ\sum_{ij} \partial_i (g^{ij} ρ\partial_j f)$ defined on a domain $Ω$ of $\mathbb R^d$ and such that there exists an orthonormal basis of $\mathcal L^2(μ)$ made of polynomials which are eigenvectors of $L$, where the polynomials are ranked according to some weighted degree. In a joint paper with D. Bakry and M. Zani this problem was solved in dimension 2 for the usual degree. In the present paper we solve it still in dimension 2, but for a weighted degree with arbitrary positive weights. |
| title | Two-dimensional diffusion orthogonal polynomials ordered by a weighted degree |
| topic | Algebraic Geometry Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2205.04949 |