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：逼近理论、几个复变量、关于复多项式的杂项问题和经典算子论。该项目最核心的问题是：在三维和更高维度的上确界范数中的调和函数最佳逼近的唯一性(二维情况最近由PI和H.S.Shapiro解决)，与幂函数的H.Bohr现象的多维类似的尖锐估计，复值调和多项式的零点的计数，C^2中Ando定理的Agler-Cole-Wermerer解释，以及最后，理解著名的双层位势算子的谱图-这一主题可以追溯到I.Fredholm的开创性工作。该项目中的大多数问题都相当简单，因此，将吸引相当多的毕业生和年轻的研究人员。这些问题也是开放的，并提出了几条新的调查途径，在远离人流较多的轨道的经典地区进行调查。在该项目中进行的研究在数学物理(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)中都有很好的应用。该项目需要本科生和研究生积极参与项目的所有阶段，并通过当地的工作室网络和教师研讨会，在对科学感兴趣的高中生中传播拟议工作中的一些更初级的非常规内容。

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.