Mathematical Sciences: Approximation Theory and Special Functions

本课题主要研究逼近和特殊函数的数学问题。后者构成了许多近似理论的基础，尽管有相当多的工作集中在分析函数本身上。要做的工作包括研究复平面上的加权位势和加权多项式，正交多项式和特殊函数的渐近性，多项式的根以及Blaschke积和有理函数的极值问题。我们还将研究Bernstein算子和Bergman核方法对数值保角映射的逼近程度。近似性是数学分析的定义特征。对物理宇宙中出现的问题的解决方案很少有简洁的表达。更常见的情况是，解必须由具有所需性质且易于计算的已知量来近似。这项研究有助于理解分析的特殊功能和发现其近似性质的双重目标。

This project focuses mathematical research on problems of approximation and special functions. The latter comprise the basis of much of approximation theory, although there is a considerable body of work concentrating on analysis of the functions themselves. Work to be done includes studies of weighted potential and weighted polynomials in the complex plane, asymptotics for orthogonal polynomials and special functions, roots of polynomials and extremal problems for Blaschke products and rational functions. Work will also be done on the degree of approximation by Bernstein operators and Bergman kernel methods for numerical conformal mapping. Approximation is the defining feature of mathematical analysis. Compact expressions of solutions of problems arising in the physical universe rarely occur. More often, the solutions must be approximated by known quantities which have desirable properties and are easily computed. This research contributes to the dual goals of understanding the special functions of analysis and discovering their approximation properties.