Computational Harmonic Analysis in Information Theory, Signal Processing, and Data Analysis

在这项研究中，研究者为信息技术、通信工程和数据分析创造了数学概念和数值方法。研究者使用伪微分算子理论、时频分析、随机矩阵理论和巴拿赫代数理论的工具，在仔细陈述的条件下，产生具有严格建立的性质的有效数值算法。本研究工作的一些具体主题是：(i)为经典和量子信息理论中的关键问题建立理论框架。具体来说，研究者考虑时变通信和量子通信中的信道容量问题；（ii） x射线晶体学、通信和雷达中的稀疏表示和压缩感知。构建非线性压缩感知框架的初步步骤；（iii）从计算效率和快速算法发展的角度出发的非交换调和分析和伪微分算子。特别关注非交换条件下算子的频谱分解及其在信号处理和无线通信中的应用。对该项目成功的强烈期望可以基于研究者在每个描述领域的现有坚实成就。在这项工作中提出的研究是前沿数学与最先进工程的几个领域的结合，寻求将抽象和应用谐波分析的先进技术以快速有效的计算方法的形式引入通信工程，信号处理和数据分析。通过将现代谐波分析方法引入工程界，这项研究活动将使重要应用的进一步进步和突破成为可能。同时，它将激发应用数学的新研究领域，并为应用数学家和工程师之间的进一步互动铺平道路。这项研究工作对整个社会的回报是多方面的，从新的信息技术能力和复杂的工具来处理今天的海量数据。通信供应商可以提高财政效率，同时为公众提供更好和更多的通信服务。其他潜在的好处包括改进医学成像和生物医学工程的方法。除了该项目广泛的技术影响之外，它还为数学/工程前沿的研究和教育提供了一种跨学科活动的模型。因此，这项研究工作有助于培养数学研究生，以发展和提高在高科技导向的社会中至关重要和迫切需要的技能。

In this research effort the investigator creates mathematical concepts and numerical methods for information technology, communications engineering, and data analysis. The investigator uses tools from pseudodifferential operator theory, time-frequency analysis, random matrix theory, and Banach algebra theory, yielding efficient numerical algorithms with rigorously-established properties under carefully stated conditions.Some concrete topics of this research effort are: (i) Development of a theoretical framework for key problems in classical and quantum information theory. Specifically, the investigator considers the channel capacity problem in time-varying communications and quantum communications;(ii) Sparse representations and compressed sensing in X-ray crystallography, communications, and radar. Initial steps toward building a framework for nonlinear compressed sensing; (iii) Noncommutative harmonic analysis and pseudodifferential operators from the point of view of computational efficiency and the development of fast algorithms.Particular attention is paid to spectral factorization for operators in a noncommutative setting, and their application in signal processing and wireless communications.Strong expectation for success of this project can be based on existing solid achievements by the investigator in each of the described areas.The research proposed in this effort is a marriage of several areas of cutting edge mathematics with state-of-the-art engineering, seeking to bring advanced techniques from abstract and applied harmonic analysis to communications engineering, signal processing, and data analysis in form of fast and efficient computational methods.By taking the modern harmonic analysis methodology into the engineering community this research activity will enable further advances and breakthroughs in important applications.At the same time it will stimulate new research areas in applied mathematics and pave the road for further interactions between applied mathematicians and engineers.The payoffs of this research effort for society at large are many, ranging from new information technology capabilities and sophisticated tools to deal with today's massive volumes of data.There are financial efficiencies to be gained by communications providers which will be accompanied by better and increased communications services for the public. Other potential benefits include improved methods for medical imaging and biomedical engineering.Beyond the project's broad technological impact, it serves as a model for the kind of cross-disciplinary activity critical for research and education at the mathematics/engineering frontier. Hence this research effort helps to train graduate students in mathematics to develop and enhance skills that are crucial and urgently needed in a high-tech oriented society.