Mathematical Sciences: Convolution Algebras and Sturm Liouville Problems

数学科学：卷积代数和 Sturm Liouville 问题

• 批准号: 9005999
• 负责人: William Connett
• 金额: $ 3万
• 依托单位: University of Missouri-Saint Louis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-07-15 至 1993-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9005999&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Convolution; Algebras; Sturm

This project seeks to blend mathematical research on the harmonic analysis of eigenfunctions of a Sturm-Liouville differential equation with the algebraic construct of a hypergroup. The hypergroup consists of an algebra of measures supported on the interval on which the equation is defined. There is a converse direction as well: a one-dimensional hypergroup defines a Sturm-Liouville problem and the characters in the measure algebra are normalized eigenfunctions of the problem. The work to be carried out falls into four areas: first, to continue exploiting the relationship between hypergroup measure algebras and the measure algebras that have Sturm-Liouville eigenfunctions as their characters. Second, investigate structural questions about hypergroups themselves. Third, the harmonic analysis of eigenfunction expansions will carried out and fourth, a student research project to develop the computational and graphic software in support of the first three areas is planned. Results of this work are expected to have applications to probability theory, orthogonal polynomials and the theory of special functions.

本项目旨在将Sturm-Liouville微分方程特征函数的调和分析的数学研究与超群的代数构造相结合。超群由在定义方程的区间上支持的测度代数组成。也有相反的方向：一维超群定义了Sturm-Liouville问题，并且度量代数中的特征是该问题的归一化特征函数。要进行的工作分为四个方面：首先，继续探索超群测度代数与具有Sturm-Liouville特征函数的测度代数之间的关系。其次，研究关于超群本身的结构性问题。第三，将进行特征函数展开的谐波分析；第四，计划一个学生研究项目来开发支持前三个领域的计算和图形软件。这项工作的结果有望在概率论、正交多项式和特殊函数理论中得到应用。