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

讨论了小波理论、非均匀采样、框架和谱瓦对偶理论中的基本问题。这些问题由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、非对易调和分析、表示论、算子代数、逼近理论和信号处理。例如，欧氏空间中单双正交小波的构造、实现和随后的理论需要所有这些学科的大量输入以及深谱瓦片结果。上述问题具有内在的数学重要性，所要表述的解具有广泛和创造性的含义，无论是对于数学还是在工程和物理中的应用。该项目的主题直接关系到磁共振成像中的快速捕获和运动问题，以及通过适当的耳蜗建模来制定压缩和降噪的算法。在量子计算和图像处理方面还有更多的应用，本项目开发的非均匀采样策略对多功能射频系统中最先进的A/D转换方法起到了一定的作用。这些依赖于现代数学分析的跨学科应用在思想的交叉培养和研究生的研究机会方面具有教育意义。

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.