Harmonic measures, polynomial inequalities, orthogonal polynomials and approximation

这是一个经典数学分析的程序，特别是在位势理论、逼近理论和正交多项式方面。一些需要考虑的问题是经典调和分析中的基本问题，因为它们涉及格林函数和调和测度的光滑性，这是理论和应用中的基本目标。主要问题是边界点附近调和测度的光滑性如何与基本区域的几何关系。这个问题的答案与逼近理论、多项式不等式和关于正交多项式的估计等问题直接相关。另一个研究方向是非经典正交多项式，它的许多问题都是由应用驱动的，而经典的工具是不可用的。特别是，Totik博士将研究关于加倍度量的正交多项式，这将打开一个新的研究方向，当经典理论(以及相关的内插理论)扩展到更一般的类时，它似乎是正确的。调和分析是信号处理、医学成像、飞机设计、模式识别、数据压缩等许多科学和工程领域的基本工具。近似理论提供了简化理论模型的方法，将复杂的函数/对象替换为更简单的函数/对象。正交多项式理论在谱理论、数值分析、静电学、统计量子力学、随机矩阵、生灭过程、预测理论、Radon变换和计算机层析成像等领域有着极其丰富的应用，这些研究将加深我们对这些领域的一些领域及其相互作用的理解。

This is a program in classical mathematical analysis, in particular in potential theory, approximation theory and orthogonal polynomials. Some of the questions to be considered are fundamental in classical harmonic analysis, since they deal with smoothness properties of Green's functions and harmonic measures, which are basic objects both in the theory and its applications. The main problem is how smoothness of harmonic measures around boundary points is related to the geometry of the underlying domain. The answer to this problem is directly linked to problems in approximation theory, polynomial inequalities and estimates on orthogonal polynomials. The other direction of research is non-classical orthogonal polynomials, for which many questions are motivated by applications, and for which classical tools are not available. In particular, Dr. Totik will study orthogonal polynomials with respect to doubling measures which will open a new direction of research that seems to be the right one when the classical theory (as well as the related interpolation theory) is extended to more general classes. The results will have direct applications to interpolation, quadrature, numerical analysis and integral equations, and so are also relevant to physics and engineering.Harmonic analysis is a fundamental tool in many branches of science and engineering like signal processing, medical imaging, airplane design, pattern recognition, data compression etc. Approximation theory provides means to simplify theoretical models via replacing complicated functions/objects by simpler ones. The theory of orthogonal polynomials is extremely rich with applications in a vast array of problems like spectral theory, numerical analysis, electrostatics, statistical quantum mechanics, random matrices, birth and death processes, prediction theory, Radon transform and computer tomography, etc. The research will enhance our understanding of some areas of these field, as well as their interaction.