FRG: Collaborative Research: Focused Research on Wavelets, Frames, and Operator Theory

FRG：协作研究：小波、框架和算子理论的重点研究

• 批准号: 0139759
• 负责人: John Benedetto
• 金额: $ 11.61万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139759&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Focused; Wavelets

FRG Collaborative ProposalPROPOSAL NUMBER PI INSTITUTION0139759 Benedetto University of Maryland, lead PIABSTRACT:Fundamental problems are addressed in wavelet theory, non-uniformsampling, frames, and the theory of spectral-tile duality. Theseproblems are inextricably interwoven by concept andtechnique. Operator theory provides the major unifying framework,combined with an integration of ideas from a diverse spectrum ofmathematics including classical Fourier analysis, noncommutativeharmonic analysis, representation theory, operator algebras,approximation theory, and signal processing. For example, theconstruction, implementation, and ensuing theory of single dyadicorthonormal wavelets in Euclidean space requires significant inputfrom all of these disciplines as well as deep spectral-tile results.There is intrinsic mathematical importance in the aforementioned problems, and the solutions to be formulated have broad and creative implications, both for mathematics and for applications in engineering and physics. The topics of this project have direct bearing on fast acquisition and motion problems in MRI, as well as in formulating algorithms for compression and noise reduction by means of proper cochlear modelling. There are furtherapplications in quantum computing and image processing, and the development of non-uniform sampling strategies by this project play a role in state of the art A/D conversion methods used in multifunction RF systems. These interdisciplinary applications depending on modern mathematical analysis have educational implications in terms of cross-fertilization of ideas and researchopportunities for graduate students.

FRG 合作提案提案编号 PI INSTITUTION0139759 马里兰州贝内代托大学，牵头 PI 摘要：小波理论、非均匀采样、框架和谱瓦对偶性理论解决了基本问题。这些问题与概念和技术密不可分地交织在一起。算子理论提供了主要的统一框架，结合了不同数学领域的思想，包括经典傅立叶分析、非交换调和分析、表示论、算子代数、逼近论和信号处理。 例如，欧几里得空间中单二正交小波的构造、实现和随后的理论需要所有这些学科的大量输入以及深入的谱图结果。上述问题具有内在的数学重要性，并且要制定的解决方案对于数学以及工程和物理中的应用都具有广泛和创造性的影响。该项目的主题与 MRI 中的快速采集和运动问题以及通过适当的耳蜗建模制定压缩和降噪算法有直接关系。在量子计算和图像处理方面有进一步的应用，并且该项目开发的非均匀采样策略在多功能射频系统中使用的最先进的A/D转换方法中发挥了作用。 这些依赖于现代数学分析的跨学科应用对于研究生的思想交叉和研究机会具有教育意义。