论文标题
在由可开发表面界定的3维域中的扩散正交多项式
Diffusion orthogonal polynomials in 3-dimensional domains bounded by developable surfaces
论文作者
论文摘要
研究了以下问题：描述三胞胎$（ω，g，μ）$，$μ=ρ\，dx $，其中$ g =（g^{ij}（x））$是与对称二阶差异差异操作员相关的（co）度量的（CO）度量 ρ\,\partial_j f)$ defined on a domain $Ω$ of $\mathbb R^n$ and such that there exists an orthonormal basis of $\mathcal L^2(μ)$ made of polynomials which are eigenvectors of $L$, and the basis is compatible with the filtration of the space of polynomials with respect to some weighted degree. 在与D. bakry和M. Zani的联合论文中，该问题在通常的程度上在维度2中解决了。在作者随后的论文中，对于任何加权程度，该问题在维度2中解决了。在本文中，在$ \partialΩ$包含一个切线开发表面的条件下，对于通常的程度，该问题在维度3中解决。该证明基于拉格尼·皮恩（Ragni Piene）给出的形式的plücker样公式。所有发现的解决方案均已概括为任何维度。
The following problem is studied: 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 $\mathbb R^n$ and such that there exists an orthonormal basis of $\mathcal L^2(μ)$ made of polynomials which are eigenvectors of $L$, and the basis is compatible with the filtration of the space of polynomials with respect 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 author's subsequent paper this problem was solved in dimension 2 for any weighted degree. In the present paper this problem is solved in dimension 3 for the usual degree under the condition that $\partialΩ$ contains a piece of a tangent developable surface. The proof is based on Plücker-like formulas in the form given by Ragni Piene. All the found solutions are generalized for any dimension.
