Mathematical Sciences: Complex Analysis: Computation and Control.

数学科学：复分析：计算与控制。

• 批准号: 8801675
• 负责人: Donald Marshall
• 金额: $ 3.78万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-15 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8801675&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Complex; Analysis; Computation

This project will focus on four areas of mathematical analysis. Work will be done on estimating two-dimensional harmonic measure which is used in potential theory and in measuring growth of analytic functions. At the present time estimates are known for the harmonic measure of curves in a disc meeting all radii in some sector. The width of the maximal sector remains to be determined. Theoretical and numerical efforts will be combined in this process. Work will continue on problems of optimization using bounded holomorphic functions. These investigations have been underway for several years. They arise from certain engineering problems in optimal control and involve finding best approximations to given functions by bounded or continuous functions defined on a circle which have analytic extensions to the interior. A third, and more recent, goal of this project will involve methods for the computation of conformal maps. There are many techniques extant for such computations and mappings of domains with smooth boundaries that generally have numerical approximants which converge rapidly. The present work is concerned with approximation involving maps of domains with corners. Recent progress has been encouraging although repetitions of the approximation process shows degradation after several passes. Efforts will be made to single out the best boundary curves for which current methods are practical. It is likely that these curves will fall into the class known as chord-arc curves which have restricted bending. Finally, in a more theoretical vein, studies will be made into the existence and nature of inner functions which are constant on Gleason parts of the maximal ideal space of a disc. While general theory indicates existence of such inner functions there is little known as to whether there are Blaschke products in this class. Related to this work is a characterization of the zero-distribution of such products. In addition to applications to potential theory, this work adds to fundamental knowledge regarding numerical approximation schemes for partial differential equations.

该项目将集中于数学分析的四个领域。 将在估计潜在理论和测量分析功能的增长中使用的二维谐波测度进行工作。 目前，估计值是符合某些部门所有半径的光盘中的曲线的谐波度量。 最大部门的宽度仍有待确定。 理论和数值努力将在此过程中合并。 使用有界的全体形态功能，将继续处理优化问题。 这些调查已经进行了几年。 它们是由最佳控制中的某些工程问题引起的，并涉及通过在圆上定义的有界或连续函数来找到最佳的近似值，这些功能对内部具有分析性扩展。 该项目的第三个也是最新的目标将涉及用于计算保形图的方法。 对于具有光滑边界的域的计算和映射，存在许多技术，这些技术通常具有迅速收敛的数值近似值。 目前的工作与涉及拐角处域的地图的近似值有关。 尽管近似过程的重复表明多次通过后，但最近的进步一直在鼓励。 将努力挑出当前方法实用的最佳边界曲线。 这些曲线可能会属于限制弯曲的和弦曲线的类。 最后，在更理论上的静脉中，将对内部功能的存在和性质进行研究，这些函数在盘的最大理想空间的Gleason部分持续不断。 虽然一般理论表明存在这种内部功能，但对于此类中是否有blaschke产品，鲜为人知。 与这项工作相关的是对此类产品的零分布的特征。 除了对潜在理论的应用外，这项工作还增加了有关部分微分方程的数值近似方案的基本知识。