生成多元多重正交多项式的一种方法
A method for generating multivariate multiple orthogonal polynomials
- Universidad de Granada（格拉纳达大学）
- Instituto de Matemáticas (IMAG)（数学研究所（IMAG））
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出扩展Koornwinder型方法，通过两种耦合方案构造多元多重正交多项式，并首次给出多种二维及高维区域上的显式公式。
中文摘要 AI 辅助
在多重正交多项式（MOPs）及其多元推广的框架内，目前仍缺乏一种统一的机制，用于在多种多维区域上生成结构化的多重正交多项式族。在本工作中，我们通过发展一种扩展的Koornwinder型方法，填补了这一空白，以构造多元多重正交多项式。我们引入了两种不同的耦合方案：将单变量MOPs族与标准正交族相结合，从而能够定义I型和II型双变量系统及其对偶双正交关系；以及将两个单变量MOPs族耦合以形成II型多元系统。利用这一通用框架，我们首次给出了多种双变量区域上多重正交多项式的显式公式，包括有界区域如抛物型区域和正方形$[0,1]^2$，非标准无界几何如正象限$(\mathbb{R}_0^+)^2$和楔形$\mathbb{V}^2$，以及三角形$T$上的新构造。此外，我们还给出了$d$维单纯形$T^d$和$d$维锥$\mathbb{V}^d$上的显式多重正交系统。
英文摘要
In the framework of multiple orthogonal polynomials (MOPs) and their extension to the multivariate setting, a unified mechanism for generating structured families of multiple orthogonal polynomials across diverse multidimensional domains is still lacking. In this work, we bridge this gap by developing an extended Koornwinder-type methodology for constructing multivariate multiple orthogonal polynomials. We introduce two distinct coupling schemes: combining a univariate MOPs family with a standard orthogonal family, which enables the definition of both Type I and Type II bivariate systems along with their dual biorthogonality relations, and coupling two univariate MOPs families to form Type II multivariate systems. Using this general framework, we provide the first explicit formulations of multiple orthogonal polynomials on a variety of bivariate regions, including bounded domains such as a parabolic domain and the square $[0,1]^2$, non-standard unbounded geometries like the positive quadrant $(\mathbb{R}_0^+)^2$ and the wedge $\mathbb{V}^2$, as well as new constructions on the triangle $T$. Furthermore, explicit multiple orthogonal systems on the $d$-dimensional simplex $T^d$ and on the $d$-dimensional cone $\mathbb{V}^d$ are given.
