Mathematical Sciences: Projects in Classical Analysis

这个数学科学研究项目小组的中心主题是近似理论。问题将从微分方程、算子插值、正交多项式和特殊函数等领域选择。一般的研究领域包括逼近理论中的势理论方法，它已经为解决逼近理论中的两个主要猜想提供了必要的工具。第二个项目涉及多元特殊函数，包括多元β和q- β型积分的求值以及在Macdonald型恒等式中寻找常数项。本文将研究经典正交多项式的多元近似的渐近展开式、积分表示和生成函数。其他的工作将集中在紧平面集上的最佳和近似的行为。具体地说，我们将努力理解最近点多项式实际上达到最小距离的集合（在一致范数中）。正交多项式在二阶差分和微分方程的谱理论中起着关键的作用。本研究将探讨一般的四阶方程。显式解和谱测度的计算将是主要目标。一个相关的研究路线将考虑多项式的根的性质相对于绝对连续的措施，在情况下，马尔可夫定理不适用。

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.