A Few Topics in Classical Analysis

提案编号：DMS-0139008 PI：Dmitry Khavinson摘要提案中描述的工作涉及经典分析中的几个主题：近似理论，几个复变量，复多项式和经典算子理论的杂项问题。该项目的核心问题是：调和函数在三维及更高维上确界范数下的最佳逼近的唯一性（二维情况最近已由PI和H. S. Shapiro解决），与H.玻尔关于幂级数的现象，复值调和多项式零点的计数，安藤定理在C ^2中的阿格勒-科尔-韦默解释，最后，理解著名的双层势算符的谱图--这个话题可以追溯到I.该项目中的大多数问题都是相当简单的陈述，因此，将吸引相当数量的研究生和年轻研究人员。这些问题也是开放式的，并建议在远离良好的轨道的古典地区进行调查的几条新途径。在该项目中进行的调查有明确的应用数学物理（势理论），应用数学（多项式，特殊函数）和数值分析（显式解决一些最重要的边值问题）。该项目要求本科生和研究生积极参与项目的各个阶段，并通过当地的教师研讨会和研讨会网络，在对科学感兴趣的高中生中传播拟议工作的一些更基本的变化。

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.