Mathematical Sciences: Studies in Approximation Theory and Orthogonal Polynomials

940577奈威这个项目涉及逼近理论、正交多项式和相关分析领域的数学研究。这项工作涉及极值问题，普通和广义多项式不等式，差分方程组和微分方程组，实、复、矩阵值三对角形(Jacobi)和带状矩阵的谱理论，Toeplitz、Hankel和Hilbert空间算子。还将继续开发用于分析正交多项式的数值方面的软件。研究的具体内容包括关于线性差分方程解的齐次和非齐次摄动解的有界性条件的研究。涉及用低次多项式加权最佳逼近单项式的极值问题的比率和比较渐近性将被学习。还将努力了解雅可比矩阵与其谱度量及其在单位圆上的类似物之间的联系。我们将把关于Christoffel函数的渐近行为的最新结果推广到复平面上的度量和广义多项式，以及它们作为逼近手段的适用性。逼近理论，特别是正交多项式理论，与许多数学分支，如微分方程组、算子论、数值积分、连分式和控制论等有着千丝万缕的联系。这些受试者在过去受到了正交多项式理论的强烈推动，而且这个过程还在以越来越大的活力继续下去。***

940577 Nevai This project is concerned with mathematical research in approximation theory, orthogonal polynomials and related areas of analysis. The work involves extremal problems, ordinary and generalized polynomial inequalities, difference and differential equations, spectral theory of real, complex and matrix-valued tridiag gonal (Jacobi) and banded matrices, Toeplitz and Hankel and Hilbert space operators. Work will also continue on the development of software directed at analyzing the numerical aspects of orthogonal polynomials. Specifics of the research include investigations into conditions on assuring boundedness for solutions of homogeneous and nonhomogeneous perturbations oflinear difference equations in terms of their unperturbed counterparts. The ratio and comparative asymptotics of extremal problems involving weighted best approximation of monomials by lower degree polynomials will be taken up. Efforts to understand the connection between a Jacobi matrix and its spectral measure and its analogues on the unit circle will also be made. Work will be done extending recent results on the asymptotic behavior of Christoffel functions to measures in the complex plane and on generalized polynomials and their suitability as means of approximation. Approximation theory and especially the theory of orthogonal polynomials has fruitful connections with many branches of mathematics such as differential equations, operator theory, numerical integration, continued fractions and control theory, to name a few. These subjects have received strong impulses from the theory of orthogonal polynomials in the past and this process continues with increasing vigor. ***