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

基本问题在小波理论，非均匀采样，帧，和理论的光谱瓦片对偶性。这些问题与概念和技术密不可分。算子理论提供了主要的统一框架，结合了来自不同数学领域的思想整合，包括经典傅立叶分析，非交换谐波分析，表示理论，算子代数，近似理论和信号处理。例如，欧几里得空间中单个二正交正交小波的构造、实现和后续理论需要来自所有这些学科以及深光谱结果的重要输入。在上述问题中有内在的数学重要性，并且要制定的解决方案具有广泛和创造性的含义，无论是对数学还是在工程和物理中的应用。本项目的主题直接关系到MRI的快速采集和运动问题，以及通过适当的耳蜗建模来制定压缩和降噪算法。在量子计算和图像处理方面有进一步的应用，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转换方法中发挥作用。这些基于现代数学分析的跨学科应用对研究生的思想交流和研究机会具有教育意义。

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.