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: 0139783
• 负责人: Gestur Olafsson
• 金额: $ 10.55万
• 依托单位: Louisiana State University
• 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=0139783&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Focused; Wavelets

FRG Collaborative ProposalPROPOSAL NUMBER PI INSTITUTION0139759 Benedetto University of Maryland, lead PI0139740 Aldroubi Vanderbilt University0139473 Jorgensen University of Iowa0139261 Heil, Wang Georgia Institute of Technology0139366 Baggett Univ. of Colorado0139783 Olafsson Lousiana State Univ.0139386 Larson Texas A&MABSTRACT: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合作性proposalpropopals number PI Institution0139759 Maryland Benedetto University，Lead PI0139740 Aldroubi Vanderbilt University0139473 Jorgensen Iowa0139261 Heil，Heil，Wang Georgia of Wang Georgia of Technology of Technology of Technology of Technology of Technology0139366 Baggett Univ。 Colorado0139783 Olafsson Lousiana State Univ.0139386 Larson Texas A＆Mabstract：在小波理论，非均匀缩采样，框架，框架和光谱瓷砖二元理论中都解决了基本问题。通过概念和技术，该问题是密不可分的。运营商理论提供了主要的统一框架，并结合了各种各样的概念的思想整合，包括经典的傅立叶分析，非强制性谐波分析，表示理论，操作员代数，近似理论和信号处理。 例如，欧几里得空间中单个二二偏二偏小波的结构，实施和随之而来的理论需要从所有这些学科以及深度的光谱范围的结果中进行大量输入。在上述问题中，要形成的解决方案和创造性的启示，并且在工程上进行了数学的影响，并且在工程上进行了数学的含义，并且在工程上进行了数学上的固有数学重要性。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算法进行压缩和降噪算法有关。量子计算和图像处理中还有进一步的应用，并且该项目的不均匀抽样策略的开发在多功能RF系统中使用的最先进的A/D转换方法中发挥了作用。 这些跨学科的应用程序取决于现代数学分析，在研究生的思想和研究范围内的交叉利用方面具有教育意义。