Collaborative Research: Bellman function, Harmonic Analysis and Operator Theory

合作研究：贝尔曼函数、调和分析和算子理论

• 批准号: 0800876
• 负责人: Serguei Treil
• 金额: $ 53.42万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800876&HistoricalAwards=false
• 关键词: Collaborative; Research; Bellman; function; Harmonic

Abstract (Volberg/Nazarov/Treil: 0758552/0800243/0800876)The proposed research is at the intersection of Geometric Measure Theory and Harmonic Analysis. The main objective of Geometric Measure Theory is to find structures in seemingly unstructured, fractal-like patterns. The classical Harmonic Analysis studies wave propagation, and investigation of singular integral operators is a crucial part of the modern approach. Our continuing research, as well as the work of several other groups of mathematicians in the US and abroad, has demonstrated that new knowledge can be obtained by exploring the interaction between these two areas. As a result of our proposal we expect to solve several important problems in Geometric Measure Theory as well as in Harmonic Analysis. The pattern recognition (i.e., problems in Geometric Measure Theory) would be advanced by using methods originating in Harmonic Analysis, and vice versa. We also expect to develop new methods to study both patterns and waves. It is expected that these newly developed techniques will have impact to adjacent areas of engineering and computer sciences such as image processing and data compression.

摘要（Volberg/Nazarov/Treil：0758552/0800243/0800876）拟议的研究是在几何测量理论与谐波分析的交集中。几何测量理论的主要目标是在看似非结构化的类似分形的模式中找到结构。经典的谐波分析研究波传播，对单数积分运算符的研究是现代方法的关键部分。我们的持续研究以及美国和国外的其他几个数学家群体的工作表明，可以通过探索这两个领域之间的相互作用来获得新知识。由于我们的建议，我们期望在几何测量理论以及谐波分析中解决一些重要问题。通过使用源自谐波分析的方法，将提出模式识别（即几何度量理论中的问题），反之亦然。我们还期望开发新的方法来研究模式和波浪。预计这些新开发的技术将对工程和计算机科学（例如图像处理和数据压缩）的相邻领域产生影响。