Analysis and Classification of Differential Equations with Orthogonal Polynomial Eigenfunctions

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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.