Problems in Function Theory

函数论问题

• 批准号: 0070782
• 负责人: John Garnett
• 金额: $ 18.37万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070782&HistoricalAwards=false
• 关键词: Problems; Function; Theory

Abstract: Garnett will work on several problems in classical one dimensionalcomplex analysis. The first problem is to approximate any Blaschke productuniformly on the open disc by Blaschke products whose zeros are sufficiently spread apart and thin that the corresponding Riesz mass isbounded in all holomorphic coordinate systems (i. e. is a Carleson measure)The approximation should be effected using explicit constructions.The second problem is to exhibit large compact sets whose complementsdo not support nonconstant bounded analytic functions. The third problemis a corona problem for infinitely connected plane domains whose boundarieslie on rectifiable Jordan curves. It too requires some new explicit constructions. The fourth problem is to show that a Jordan curve depends continuously on its welding map, which is the correspondence on theunit circle that connects the conformal mapping of the inside of thedisc to one side of the curve and the outside of the disc to the other side of the curve.The methods to be used on these problems will be constructive so that theycan be give explicit computer aided constructions of analytic functionsand conformal mappings. Analytic functions and conformal mappings havebroad applications in fluid dynamics, acoustics, and electrical engineering,and in these applications constructions are more useful than general existence theorems.

翻译后摘要：加内特将在经典的一维复分析的几个问题。 第一个问题是用Blaschke乘积一致逼近开圆盘上的任何Blaschke乘积，这些Blaschke乘积的零点足够分散和薄，使得相应的Riesz质量在所有全纯坐标系中是有界的（即，e.是Carleson测度）的逼近应该使用显式构造来实现。第二个问题是展示大的紧集，其补集不支持非常数有界解析函数。 第三个问题是边界位于可求长Jordan曲线上的无限连通平面区域的冠问题。 它也需要一些新的明确的结构。 第四个问题是证明Jordan曲线连续依赖于其焊接图，这是单位圆上的对应关系，它把圆的内侧到曲线的一侧，圆的外侧到曲线的另一侧的保角映射连接起来。在这些问题上所用的方法将是构造性的，以便能给出解析函数和保角函数的明确的计算机辅助构造。映射。 解析函数和共形映射在流体力学、声学和电气工程中有着广泛的应用，在这些应用中，构造比一般的存在定理更有用。