Analysis and Classification of Differential Equations with Orthogonal Polynomial Eigenfunctions

PI将研究几个关于分析和分类正交多项式到谱型微分方程式的问题。这些问题中最一般的是所谓的BKS(N，M)问题，它寻求将所有N阶常微分方程组分类，直到变量的实线性变化，即与涉及M阶导数函数的Sobolv双线性形式正交的多项式特征函数序列。当M=0时，这个问题是经典的，得到了Bochner和H.L.Krall的关键结果。最近，Littlejohn，K.H.Kuan和D.W.Lee在M=1和M=2情形下的二阶方程方面取得了新的进展。这些问题在正交多项式理论和微分方程组理论中是经典的，自1929年以来一直是数学分析的一个突出挑战。这些问题在数学、应用数学、物理和工程的许多领域都有广泛的应用。事实上，具有正交多项式解的经典二阶方程在包括量子力学在内的许多应用数学和物理领域中都是重要的。此外，BKS(N，M)问题的完整解决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。高阶算例的应用前景看好。事实上，我们发现的每一个算例都将对采样和内插理论的应用以及在工程(通信和信号处理理论)中的应用产生影响。世界各地的数学同行在理论和技术上的最新进步，为这些困难的问题很快就会产生全球解决方案带来了希望。

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.