Christoffel functions and applications

这是一个为期三年的计划，在经典分析，特别是在潜在的理论与应用在近似理论和正交多项式。一个共同的统一主题是Christoffel函数和再生核的行为及其各种应用。另一个统一的主题是所谓的多项式逆图像方法。应用包括正交多项式的精细零行为（直接翻译为Jacobi矩阵的特征值），随机矩阵理论/统计物理和各种多项式不等式的普遍性结果。其他问题涉及到非经典正交多项式（关于加倍权重）和近似齐次多项式及其水平集。在其中几个问题中，多项式逆像方法--过去已经产生了尖锐和重要的结果--将发挥至关重要的作用，作为回报，将要开发的工具将帮助我们更好地理解这种强大的技术。研究的核心将是在近似理论和正交多项式的相对遥远的领域中系统地使用势理论和经典调和分析。正交多项式的各种性质的拟议研究旨在通过引入新的工具来推进一个非常经典的数学领域（可以追溯到200多年前），该领域具有众多的联系和应用。这些结果也与数学、物理和工程的其他分支有关。这项研究将激发学生的兴趣，并为他们改善研究环境。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。部分研究成果和方法将被整合到相关课程中，从而促进K-12科学和数学教师的专业发展。将特别重视与公众的外联。这将通过出版教育文章来实现，从而提供对科学的理解和欣赏。这些教育材料，正如研究结果一样，将广泛分发。讲座将在不同层次举行，从本科社团到专业会议。

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.