Mathematical Sciences: Symmetry and Orthogonal Polynomials

数学科学：对称性和正交多项式

• 批准号: 8802400
• 负责人: Charles Dunkl
• 金额: $ 7.51万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-05-01 至 1992-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8802400&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Symmetry; Orthogonal; Polynomials

Several areas within the theory of special functions will be investigated. Particular emphasis will be placed on the development of a mathematical theory of orthogonal polynomials of several variables in which there will be a concentration on functions defined on the n-sphere. These polynomials are taken to be orthogonal with respect to a weight function which is invariant under a finite reflection group. The weight functions are powers of products of linear functions whose zero sets are the mirrors for the reflections in the group. A form of Selberg's integral, corresponding to the octahedral group, is an important special case. By means of a commutative algebra of first-order differential-difference operators, which have an action analogous to that of partial derivatives in the theory of spherical harmonics, work will focus on three problems. The first is to construct a nonabelian fractional integral operator which intertwines ordinary differentiation and the differential- difference operators. This gives a mapping from harmonic functions and other eigenfunctions of the Laplacian to the functions under investigation. Secondly, by use of these operators and their adjoints, work will be done constructing explicit orthogonal bases and norms for functions on the sphere. Emphasis will be placed first on three and four dimensional examples. Thirdly, work will be done extending a recent positivity result on the Poisson kernel for the ball to finding this kernel explicitly for the general reflection group. This work is expected to have application to theoretical physics. It has already demonstrated its utility by playing a major role in the grinding of mirrors for the new Keck 10-meter telescope.

特殊函数理论中的几个领域将在 研究了 将特别强调 正交多项式的数学理论的发展 几个变量，其中将有一个集中在 在n-sphere上定义的函数。 这些多项式是 相对于权重函数是正交的， 在有限反射群下不变。 权函数 是线性函数的乘积的幂，其零点集是 反射的镜子。 的一种形式 Selberg积分，对应于八面体群，是一个 重要的特殊情况。 借助于一阶交换代数 微分-差分算子，其作用类似于 球面理论中的偏导数 谐波，工作将集中在三个问题。 一是 构造一个非交换分数次积分算子， 普通的微分和微分交织在一起 差分算子 这给出了一个从调和 拉普拉斯算子的函数和其他本征函数 正在调查的职能。 第二，利用这些 运算符及其伴随，工作将完成构造 球面上函数的显式正交基和范数。 重点将首先放在三维和四维 例子. 第三，将开展工作， 关于Poisson核的正性结果 这个核显式地用于一般反射群。 这项工作有望应用于理论 物理学 它已经通过播放一个 在新凯克10米镜的研磨中发挥了重要作用 望远镜