Mathematical Sciences: Hypergroups, Measure Algebras and Polynomials in Rn

这个项目的重点是广义调和分析 涉及高维集合上的正交函数的 欧几里得空间 研究的新特点是， 审议中的各套并不构成一个集团。 这种缺乏 代数结构通常可以通过存在 在测度代数中， 支持的设置。 这里考虑的结构 被称为超群，其中代数运算 就是卷积 如果正交基的元素 函数充当超群的同态，则得到一个 可行的数学结构 例如，最初的一个- 维工作在该地区使用的特征函数的斯特姆- Liouville问题为基础。 本项目旨在 推广了已有的紧一维结构定理 超群到非紧的情形。 其他工作将侧重于 刻画高等紧支超群 尺寸，开发磁盘的谐波分析 多项式和球面波函数，并分析了 特殊非超群测度族结构 代数 超群的概念虽然抽象，但它的根源在于 概率论、正交多项式和 特殊功能。 具体应用包括显式 环形球状波函数的表示，即使 函数不能以封闭的形式给出， 特征函数的超群解释 概率论

The focus of this project is a generalized harmonic analysis involving orthogonal functions on sets in higher dimensional Euclidean space. The novel feature of the research is that the sets under consideration do not form a group. This lack of algebraic structure can often be compensated for by the existence of an adequate algebraic structure in the measure algebra supported on the set. The structures under consideration here are referred to as hypergroups in which the algebraic operation is convolution. If the elements of a basis of orthogonal functions act as homomorphisms of the hypergroup, then one gets a viable mathematical structure. For example, the initial one- dimensional work in the area used eigenfunctions of a Sturm- Liouville problem as a basis. The current project seeks to extend existing structure theorems for compact one-dimensional hypergroups to the non-compact case. Other work will focus on characterizing the compactly supported hypergroups in higher dimensions, developing a harmonic analysis of the disk polynomials and the spheroidal wave functions and analyzing the structure of specific families of non-hypergroup measure algebras. The hypergroup concept, abstract as it is, has its roots in probability theory, orthogonal polynomials and the theory of special functions. Particular applications include the explicit representations for annular spheroidal wave functions even though the functions cannot be given in closed form and the development of hypergroup interpretations of characteristic functions in probability theory.