Harmonic measures in approximation and orthogonal polynomials

这是一个为期三年的经典分析项目，特别侧重于近似理论和正交多项式。一个共同的统一主题是调和措施的外观在各种问题进行调查，以及使用快速减少（Dirac-delta-like）多项式合并局部逼近，以创建全球性的。应用包括随机矩阵理论的普遍性结果，各种多项式不等式，和基于Christoffel函数的域重建过程。 该项目的其他部分涉及非经典正交多项式/Christoffel函数，多项式逼近在几个变量，并开发新的工具（基于快速减少多项式），为他们的研究。 智力上的优点在于实现了更好地理解谐波措施的作用，在近似理论和正交多项式，以及在表现的有用性，快速减少多项式在各个领域的数学有点远离主要议题的项目。例如，在成功完成该项目后，主要研究者将对一般凸域上的多项式逼近率进行表征（就平滑度而言），这将关闭半个多世纪以来的经典研究路线。将应用的技术将包括系统使用谐波分析近似理论，从而提供新的工具，在后者的领域。 所提出的研究将有更广泛的影响，超越近似理论和正交多项式，因为结果是相关的数学，物理和工程的其他分支。例如，各种形式的所谓普适性猜想都起源于理论和统计物理学的研究，这些研究表明了某种普适的局部行为，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，该提案提供了一种工具来研究这种精细行为，并在一般条件下严格证明普遍性。拟议的正交多项式研究具有更广泛影响的另一个领域是根据扫描数据（MRI型扫描）进行域/形状重建。 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域没有孔/腔（可能不真实或可能事先不知道的信息）。本建议奠定了理论基础的修改方法适用于域可能有孔。通过相对简单的迭代，该算法不仅可以重建域的外边界（原始目标），而且实际上还可以重建内部孔。研究的更广泛影响还包括激发本科生和数学专业学生的兴趣，并为他们改善研究环境。研究生和博士学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。将特别重视对公众的宣传。这将通过发表与研究有关的教育文章来实现，从而提供对科学的理解和欣赏。相关讲座将在不同层次举行，从本科生社团到专业会议。

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.