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: 0758552
• 负责人: Alexander Volberg
• 金额: $ 61.59万
• 依托单位: Michigan State 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 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0758552&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.

（Volberg/Nazarov/Treil: 0758552/0800243/0800876）本文的研究方向是几何测量理论和谐波分析的交叉领域。几何测量理论的主要目标是在看似无结构的分形模式中找到结构。经典的调和分析研究的是波的传播，而奇异积分算子的研究是现代方法的重要组成部分。我们的持续研究，以及美国和国外其他几组数学家的工作表明，通过探索这两个领域之间的相互作用，可以获得新的知识。由于我们的建议，我们期望解决几何测量理论和谐波分析中的几个重要问题。模式识别（即几何测量理论中的问题）将通过使用谐波分析的方法来推进，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图像处理和数据压缩等工程和计算机科学的邻近领域产生影响。