Harmonic measures in approximation and orthogonal polynomials

这是一个为期三年的经典分析项目，重点是近似理论和正交多项式。一个共同的统一主题是在各种待研究的问题中谐波测度的出现，以及使用速降（狄拉克- δ样）多项式来合并局部近似，从而创建全局近似。应用包括随机矩阵理论中的普适性结果、各种多项式不等式和基于Christoffel函数的域重构程序。项目的其他部分涉及非经典正交多项式/克里斯托费尔函数，多个变量的多项式近似，以及为他们的研究开发新的工具（基于快速递减多项式）。智力上的优点在于更好地理解谐波测度在近似理论和正交多项式中的作用，以及在与项目主要主题有些距离的数学各个领域中速降多项式的有用性的表现。例如，在项目成功完成后，首席研究员将对一般凸域上的多项式逼近率进行表征（就平滑度而言），这将结束半个多世纪以来的经典研究路线。所应用的技术将包括在近似理论中系统地使用谐波分析，从而为后一个领域提供新的工具。所提出的研究将具有比近似理论和正交多项式更广泛的影响，因为其结果与数学，物理和工程的其他分支相关。例如，各种形式的所谓普适性猜想起源于理论物理学和统计物理学的研究，这些研究表明了某种普遍的局部行为，这种行为与描述给定系统的原始量无关。在某种形式下，数学公式转化为正交多项式的精细局部行为，该建议提供了研究这种精细行为和在一般条件下严格证明通用性的工具。另一个提出的正交多项式研究具有更广泛影响的领域是扫描数据的域/形状重建（mri型扫描）。最近的一种方法是基于面积生成的正交多项式，但该方法假设要重建的区域没有孔/腔（可能不真实或可能事先不知道的信息）。本文的提出为该方法的改进奠定了理论基础，使其适用于可能存在空穴的区域。通过相对简单的迭代，该算法不仅可以重建域的外部边界（原始目标），而且还可以重建内部孔。更广泛的影响还包括激发本科生和数学专业的兴趣，改善他们的研究环境。研究生和博士生将有机会学习几个不同学科的基本成果和技术，以及它们之间的相互关系。将特别强调与一般公众的接触。这将通过发表与研究相关的教育文章来实现，从而提供对科学的理解和欣赏。相关讲座将在不同级别举行，从本科学会到专业会议。

This is a three-year project in classical analysis that focuses, in particular, on approximation theory and orthogonal polynomials. A common unifying theme is the appearance of harmonic measures in various problems to be investigated, as well as the use of fast decreasing (Dirac-delta-like) polynomials to merge local approximants so as to create global ones. The applications include universality results in random matrix theory, various polynomial inequalities, and a domain reconstruction procedure based on Christoffel functions. Other parts of the project are related to nonclassical orthogonal polynomials/Christoffel functions, to polynomial approximation in several variables, and to developing new tools (based on fast decreasing polynomials) for their study. The intellectual merit lies in achieving a better understanding of the role of harmonic measures in approximation theory and orthogonal polynomials, as well as in the manifestation of the usefulness of fast decreasing polynomials in various areas of mathematics somewhat remote from the main topic of the project. For example, upon successful completion of the project the Principal Investigator will have a characterization (in terms of smoothness) of the rate of polynomial approximation on general convex domains, which will close a classical line of research going back more than half a century. The techniques to be applied will include systematic usage of harmonic analysis in approximation theory, thereby offering new tools in the latter field. The proposed research will have broader impact beyond approximation theory and orthogonal polynomials, for the results are relevant to other branches of mathematics, physics, and engineering. For example, various forms of the so-called universality conjecture have originated from studies in theoretical and statistical physics, studies that showed a certain universal local behavior that was independent of the original quantities describing the given system. In some form the mathematical formulation translates into a fine local behavior of orthogonal polynomials, and the proposal offers a tool to study this fine behavior and to prove rigorously universality under general conditions. Another area where the proposed research on orthogonal polynomials has broader impact is domain/shape reconstruction from scanned data (MRI-type scans). A recent method is based on area-generated orthogonal polynomials, but that method assumed that the domain to be reconstructed was without holes/cavities (information that may not be true or may not be known in advance). The present proposal lays the theoretical foundation for a modification of the method applying to domains that may have holes. By a relatively simple iteration, the algorithm will not only reconstruct the outer boundary of the domain (the original goal), but it will actually allow the reconstruction of the inner holes as well. Broader impacts of the research also include stimulating interest in undergraduates and mathematics majors and enhancing the research environment for them. Graduate and Ph.D. students will have the opportunity to learn the basic results and techniques of several different disciplines, as well as their interrelations. Special emphasis will be made in outreach to the general public. This will be achieved by publishing educational articles connected with the research, thereby offering understanding and appreciation for science. Relevant lectures will be held at different levels from undergraduate societies to professionals meetings.