Mathematical Sciences: Problems in Function Theory

数学科学：函数论问题

• 批准号: 9401269
• 负责人: John Garnett
• 金额: $ 13.48万
• 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: 1994
• 资助国家: 美国
• 起止时间: 1994-04-15 至 1998-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401269&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Function; Theory

Garnett 9401269 The research supported by this award focuses on problems arising in the mathematical theory of functions of one complex variable and one-dimensional Schrodinger operators. The work continues investigations in four problem areas. In the first, work will be done investigating the density of Blaschke products whose roots are restricted to interpolating sequences for the algebra of bounded analytic functions. It is expected that all Blaschke products are uniform limits of the restricted ones and that the closed convex hull of these Blaschke products is the unit ball for the algebra. A second line of investigation concerns analytic capacity to help understand the nature of sets having positive Hausdorff dimension but which have zero capacity. Work will also be done on determining for which powers the derivative of a conformal mapping is integrable along straight lines in simply connected domains. Related investigations will consider the same question when the integrals are taken with respect to area measure. Efforts to understand the gap sequences (bands) in one-dimensional periodic Schrodinger operators will be taken up for operators with complex potentials. Recent work has shown that the spectrum can be realized in terms of Riemann surfaces. The purpose of this work is to obtain a more intrinsic description of the spectrum. Complex function theory is the study of the analytic and geometric properties of differentiable functions of a complex variable. The subject has grown in time to include ancillary lines of investigation such as quasiconformal mapping, quasiregular mapping and potential theory. Work in the field is motivated by problems in geometry, differential equations, fluid dynamics and control theory. Recent work also involves studies of computational conformal mapping through new applications of circle packing. ***

加内特9401269 该奖项支持的研究重点是在一个复杂的变量和一维薛定谔算子的函数的数学理论中出现的问题。 这项工作继续在四个问题领域进行调查。 第一部分研究了有界解析函数代数中根被限制在插值序列上的Blaschke积的稠密性。 期望所有的Blaschke积都是约束积的一致极限，并且这些Blaschke积的闭凸船体是代数的单位球. 第二条线的调查关注的分析能力，以帮助理解集的性质具有积极的豪斯多夫维数，但有零的能力。 工作也将完成确定哪些权力的衍生物的共形映射是可积的沿着直线在简单连通域。 有关的研究在对面积测度进行积分时也会考虑同样的问题。 努力了解差距序列（带）在一维周期薛定谔算子将采取了复杂的潜力运营商。 最近的工作表明，频谱可以实现的黎曼曲面。 这项工作的目的是获得一个更内在的描述的频谱。 复变函数论是研究复变函数的解析和几何性质的学科。 这门学科随着时间的推移而发展，包括了一些辅助的研究领域，如拟共形映射、拟正则映射和势理论。 在该领域的工作是由几何问题，微分方程，流体动力学和控制理论的动机。 最近的工作还涉及研究计算保角映射通过新的应用程序的圆包装。 ***