Mathematical Sciences: Studies in Orthogonal Polynomials and Approximation Theory

内瓦伊计划继续他在逼近理论、正交多项式和相关分析领域的研究，这些领域涉及各种极值问题、普通和广义多项式不等式、差分方程组、实值、复值和矩阵值的Jacobi、Hessenberg和带状矩阵的谱理论、Toeplitz和Hankel形式，以及Hilbert空间算子。此外，他还将继续致力于他在MATHEMICATA中的“正交多项式”软件项目，以及正交多项式的数值方面的工作。他的主要研究将集中在三个领域：单位圆和单位圆弧上的正交多项式，广义多项式和多项式不等式，以及线性差分方程组及其解的增长性。逼近理论和正交多项式构成了数学分析的一个重要部分，因为(I)它们为在实际生活中应用“纯”数学中的各种结果提供了理论基础，(Ii)它们在理论和实践之间建立了一座天然的桥梁。正交多项式的特殊用途源于这样一个事实：(I)它们很容易通过稳定的三项递归公式计算，(Ii)它们是将“一般”函数展开成良好级数的自然媒介，以及(Iii)它们的零点特别适合于插补和求积过程。求积过程使人们能够以高精度计算涉及积分的非常复杂的表达式。这一建议的主要主题，即Szego的正交多项式理论的扩展，对于这些目的特别有用。作者希望找到具有坚实理论基础的有效方法。

Nevai Abstract Nevai plans to continue his research in approximation theory, orthogonal polynomials, and related areas of analysis involving various extremal problems, ordinary and generalized polynomial inequalities, difference and differential equations, spectral theory of real, complex, and matrix-valued Jacobi, Hessenberg, and banded matrices, Toeplitz and Hankel forms, and Hilbert space operators. In addition, he will continue working on his "Orthogonal Polynomials" software project in Mathematica and on numerical aspects of orthogonal polynomials. The primary focus of his research will be concentrated on three areas: orthogonal polynomials on the unit circle and on arcs of the unit circle, generalized polynomials and polynomial inequalities, and linear difference equations and growth of their solutions. Approximation theory and orthogonal polynomials form an essential part of mathematical analysis in the sense that (i) they provide theoretical foundations for real life applications of various results in "pure" mathematics, and that (ii) they yield a natural bridge between theory and practice. The extraordinary usefulness of orthogonal polynomials stems from the facts that among others (i) they are easily computable by a stable three term recursion formula, (ii) they are a natural medium for expanding "general" functions into well behaved series, and that (iii) their zeros are especially suitable for interplation and quadrature processes. Quadrature processes enable one to evaluate very complicated expressions involving integrals with high degree of precision. The primary subject of this proposal, that is, extensions of Szego's theory of orthogonal polynomials, is especially useful for theses purposes. The proposer hopes to find efficient methods with solid theoretical foundations.