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

在小波理论，非均匀采样，帧，和频谱瓦片对偶理论的基本问题得到解决。这些问题是由概念和技术不可分割地交织在一起。算子理论提供了主要的统一框架，结合了各种数学思想，包括经典傅立叶分析、非对易调和分析、表示论、算子代数、近似理论和信号处理。 例如，欧氏空间中单二阶正交小波的构造、实现和随后的理论需要来自所有这些学科的重要投入以及深刻的频谱瓦片结果。上述问题具有内在的数学重要性，所要制定的解决方案对数学以及工程和物理应用都具有广泛的创造性意义。该项目的主题直接关系到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。在量子计算和图像处理中有进一步的应用，本项目开发的非均匀采样策略在多功能RF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在研究生的思想交流和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.