A Few Topics in Classical Analysis

提案编号：DMS-0139008PI：Dmitry Khavinson 摘要提案中描述的工作涉及经典分析中的几个主题：逼近论、几个复变量、复多项式的各种问题和经典算子理论。该项目最核心的问题是：在三维及更高维度的最高范数中调和函数的最佳近似的唯一性（二维情况最近已由 PI 和 H.S.Shapiro 解决）、与 H. Bohr 幂级数现象的多维类似物相关的敏锐估计、计算复值调和多项式的零、Agler-Cole-Wermer 对 C^2 中 Ando 定理的解释以及，最后，了解著名的双层势算子的光谱图——这个话题可以追溯到 I. Fredholm 的开创性工作。该项目中的大多数问题都相当简单地陈述，因此，将吸引相当多的研究生和年轻研究人员。这些问题也是开放式的，并提出了在这些远离交通路线的经典地区进行调查的几种新路径。该项目中进行的研究对数学物理（势论）、应用数学（多项式、特殊函数）和数值分析（一些最重要的边值问题的显式解）有明确的应用。该项目需要本科生和研究生积极参与项目的各个阶段，并通过当地的教师讲习班和研讨会网络，向对科学感兴趣的高中生传播拟议工作中的一些更基本的另类内容。

Proposal Number: DMS-0139008PI: Dmitry KhavinsonABSTRACTThe work described in the proposal concerns several topicsin Classical Analysis: approximation theory, several complexvariables, miscellaneous problems on complex polynomials andclassical operator theory. The most central questions for theproject are: uniqueness of best approximation by harmonicfunctions in the supremum norm in dimensions three and higher(the two-dimensional case has been recently settled by the PIand H.S.Shapiro), sharp estimates relating to multidimensionalanalogs of H. Bohr's phenomenon for power series, countingzeros of complex-valued harmonic polynomials, Agler-Cole-Wermerinterpretation of Ando's theorem in C^2 and, finally,understanding the spectral picture of the celebrated doublelayer potential operator -a topic going back to the pioneeringwork of I. Fredholm.Most of the questions in the project are rather simply statedand, accordingly, will attract a fair number of graduatestudents and young researchers. The problems are also open-endedand suggest several new paths of investigation in theseclassical areas away from the well traveled tracks. Theinvestigations undertaken in the project have well-definedapplications to Mathematical Physics (Potential Theory),Applied Mathematics (polynomials, special functions) andNumerical Analysis (explicit solutions of some of the mostimportant boundary value problems). The project entailsactive involvement of undergraduate and graduate students atall stages of the project and, via local network of workshopsand seminars for teachers, to disseminate some of the moreelementary off-beats of the proposed work among high schoolstudents interested in science.