Harmonic measures, polynomial inequalities, orthogonal polynomials and approximation

这是一个经典数学分析的程序，特别是势理论，近似理论和正交多项式。一些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平滑性，这是理论和应用的基本对象。主要问题是边界点周围谐波测度的平滑度如何与底层域的几何形状相关。这个问题的答案与近似理论、多项式不等式和正交多项式估计中的问题直接相关。另一个研究方向是非经典正交多项式，它的许多问题是由应用驱动的，并且经典工具不可用。特别是，Totik博士将研究正交多项式的加倍措施，这将开辟一个新的研究方向，似乎是一个正确的，当经典理论（以及相关的插值理论）扩展到更一般的类。其结果将直接应用于插值、正交、数值分析和积分方程，因此也与物理和工程有关。谐波分析是信号处理、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。近似理论提供了通过用更简单的函数/对象代替复杂的函数/对象来简化理论模型的方法。正交多项式理论在光谱理论、数值分析、静电学、统计量子力学、随机矩阵、生死过程、预测理论、氡变换和计算机断层扫描等领域有着极其丰富的应用。该研究将加深我们对这些领域的某些领域以及它们之间的相互作用的理解。

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.