Christoffel functions and applications

这是一个三年的课程，学习经典分析，特别是位势理论，以及在逼近理论和正交多项式中的应用。一个共同的统一主题是Christoffel函数和复制内核及其各种应用程序的行为。另一个统一的主题是所谓的多项式逆像方法。这些应用包括正交多项式的精细零性态(直接转化为雅可比矩阵的特征值)、随机矩阵理论/统计物理中的普适性结果以及各种多项式不等式。其他问题涉及非经典的正交多项式(关于加倍的权重)和齐次多项式及其水平集的逼近。在其中的几个问题中，多项式反映象方法--过去已经产生了尖锐而显著的结果--将发挥关键作用，而作为回报，将要开发的工具将帮助我们更好地理解这一强大的技术。研究的核心将是在相对较远的逼近理论和正交多项式领域中系统地使用位势理论和经典调和分析。提出的研究正交多项式的各种性质的目的是通过引入这些新工具来推进一个具有众多联系和应用的非常经典的数学领域(可追溯到200多年前)。这些结果也适用于数学、物理和工程学的其他分支。这项研究将激发学生的兴趣，改善他们的研究环境。研究生和博士生将有机会学习不同学科的基础知识和强大的技术，以及它们之间的相互关系。一些成果和方法将被整合到相关课程中，这反过来可能会促进K-12科学和数学教师的专业发展。活动的重点将特别放在外展一般市民身上。这将通过发表教育文章来实现，从而提供对科学的理解和欣赏。这些教育材料，就像研究结果一样，将被广泛分发。讲座将在不同层次举行，从本科生社团到专业人士会议。

This is a three year program in classical analysis, in particular in potential theory with applications in approximation theory and orthogonal polynomials. A common unifying theme is the behavior of Christoffel functions and reproducing kernels and their various applications. Another unifying theme is the so called polynomial inverse image method. The applications include fine zero behavior of orthogonal polynomials (with direct translation to eigenvalues of Jacobi matrices), universality results in random matrix theory/statistical physics and various polynomial inequalities. Other questions are related to non-classical orthogonal polynomials (with respect to doubling weights) and approximation by homogeneous polynomials and by their level sets. In several of these problems the polynomial inverse image method - that has already produced sharp and significant results in the past - will play a crucial role, and, in return, the tools to be developed will help us in better understanding this powerful technique. At the heart of the research will be the systematic usage of potential theory and classical harmonic analysis in the relatively distant areas of approximation theory and orthogonal polynomials.The proposed study of various properties of orthogonal polynomials is aimed, by introducing there new tools, to advance a very classical field in mathematics (going back to over 200 years) which has multitudes of connections and applications. The results are relevant to other branches of mathematics, physics and engineering, as well. The research will stimulate interest in students and enhance research environment for them. Graduate and PhD students will have the opportunity to learn the fundamentals and powerful techniques of different disciplines, as well as their interrelations. Some of the results and methods will be integrated into related courses, which in turn may advance the professional development of K-12 science and mathematics teachers. Special emphasis will be made to outreach general public. This will be achieved by publishing educational articles, thereby offering understanding and appreciation for science. These educational materials, just as the findings of the research, will be widely distributed. Lectures will be held at different levels from undergraduate societies to professionals meetings.