Mathematical Sciences: Projects in Classical Analysis

数学科学研究的中心主题 项目组是近似理论。 将选择问题 从微分方程，插值， 算子、正交多项式和特殊函数。 研究的一般领域包括潜在的理论 近似理论中的方法已经提供了 解决两个主要问题的必要工具， 近似理论 第二个项目涉及多变量 特殊功能，包括多变量beta的评估 和q-β型积分及Macdonald中常数项的求法 类型的身份。 渐近展开，积分 多元类比的表示和母函数 的经典正交多项式将进行研究。 其他 工作将集中在最佳和接近最佳的行为 紧平面集上的逼近 具体而言，努力将 理解最近点多项式的集合 实际上实现最小距离（在统一规范中）。 正交多项式在谱分析中起着关键作用 二阶差分和微分方程理论。 本研究将探讨一般四阶 方程 显式解和计算 光谱测量将是主要目标。 一条相关的线 研究将考虑多项式根的性质 关于绝对连续测度的正交 马尔可夫定理不适用的情况。

The central theme of this mathematical science research project group is approximation theory. Problems will be selected from areas such as differential equations, interpolation of operators, orthogonal polynomials and special functions. General areas of research include potential theoretic methods in approximation theory which has already provided the necessary tools for the resolution of two major conjectures in approximation theory. A second project concerns multivariate special functions including the evaluation of multivariate beta and q-beta type integrals and finding constant terms in Macdonald type identities. Asymptotic expansions, integral representations and generating functions of multivariate analogs of the classical orthogonal polynomials will be studied. Other work will focus on the behavior of best and near best approximants on compact plane sets. Specifically, efforts will be made to understand the sets where nearest point polynomials actually achieve the minimal distance (in the uniform norm.) Orthogonal polynomials play a key role in the spectral theory of second order difference and differential equations. This research will investigate the general fourth order equations. Explicit solutions and the computation of the spectral measure will be primary goals. A related line of investigation will consider the nature of roots of polynomials orthogonal with respect to absolutely continuous measures in cases where Markov's theorem does not apply.