Analysis and Classification of Differential Equations with Orthogonal Polynomial Eigenfunctions

正交多项式本征函数微分方程的分析与分类

• 批准号: 9970478
• 负责人: Lance Littlejohn
• 金额: $ 5.52万
• 依托单位: Utah State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-15 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970478&HistoricalAwards=false
• 关键词: Analysis; Classification; Differential; Equations; Orthogonal

The PI will investigate several problems regarding the analysis andclassification of orthogonal polynomials to spectral-type differentialequations. The most general of these problems is the so-calledBKS(N,M) problem which seeks a classification of all ordinary differentialequations of integer order N, up to a real linear change of variable, thathas a sequence of polynomial eigenfunctions which are orthogonal withrespect to a Sobolev bilinear form involving the M-th derivative offunctions. When M=0, this problem is classical with key results due toBochner and H. L. Krall. Recent progress by Littlejohn, K. H. Kwon and D. W.Lee has been made on second-order equations in the cases M=1 and M=2.These problems in the theory of orthogonal polynomials and differentialequations are classical and have been an outstanding challenge tomathematical analysis since 1929. Applications of these problems abound inmany areas of mathematics, applied mathematics, physics, and engineering.Indeed, the classical second-order equations having orthogonal polynomialsolutions are important in many areas of applied mathematics and physics,including quantum mechanics. Moreover, a complete solution to the BKS(N,M)problem will require an indepth knowledge of distributions, differenceequations, moment theory, Lie algebras, complex analysis, and operatortheory. The applications of the higher-order examples look very promising.Indeed, each of the examples that we find will have an impact on appliedsampling and interpolation theory with applications to engineering (theoryof communication and signal processing). Recent improvements in both theoryand techniques, by mathematical colleagues throughout the world, lend hopethat these difficult problems will soon yield global solutions.

PI将研究关于谱型微分方程的正交多项式的分析和分类的几个问题。这些问题中最一般的是所谓的BKS（N，M）问题，它寻求对所有整数阶N的常微分方程的分类，直到变量的真实的线性变化，即一个多项式特征函数序列，这些多项式特征函数关于涉及M阶导数函数的Sobolev双线性型是正交的。当M=0时，这个问题是经典的，其关键结果是Bochner和H. L.克劳作者：Littlejohn，K. H. Kwon和D. W.Lee在M=1和M= 2的情形下对二阶方程作了研究。这些问题是正交多项式和微分方程理论中的经典问题，自1929年以来一直是数学分析中的一个突出挑战。这些问题在数学、应用数学、物理学和工程学的许多领域都有广泛的应用。实际上，具有正交多项式解的经典二阶方程在应用数学和物理学的许多领域都很重要，包括量子力学。此外，BKS（N，M）问题的完整解决方案将需要深入了解分布，差分方程，矩理论，李代数，复分析和算子理论。高阶例子的应用看起来非常有前途。事实上，我们发现的每个例子都将对应用采样和插值理论产生影响，并将其应用于工程（通信和信号处理理论）。世界各地的数学界同仁最近在理论和技术方面的改进使我们相信，这些难题很快就会有全局性的解决办法。