Time-Frequency and Applied Harmonic Analysis

HeilDMS-0806532 本课题主要有四个研究方向：（a）时频独立性理论及相关问题：（B）非交换相干态局域化的量化 （c）仿射伪微分算子的应用 （d）剪切波变换的多小波发展， 这些相互关联的主题中的每一个都包含了与信号的冗余表示和转换中的基本问题直接相关的深层次问题。 每个主题都涉及到数学分析的广泛领域，并且需要从抽象分析和应用调和分析中综合不同的思想，包括伪微分算子理论，非交换Banach代数，以及适用于二维和更高信号处理的多分辨率分析的新形式。 在这个项目中解决的问题涉及到现代信号处理的基础数学工具的基本问题，包括无线通信，雷达，地震传感和图像处理等应用。这些主题的进展产生了对现实世界的科学和工程计划具有潜在影响的适用数学。 这项工作的所有方面都与教育工作密切相关，无论是在研究生还是本科生水平。 研究者的学生直接合作，在本研究中，接收知识和经验的广度和深度，这将促进数学，物理和工程之间的跨学科互动的持续承诺。 与该项目有关的主题和项目问题的进展被纳入调查员指导的课程和研讨会。 这位博士继续着他在培养研究生、与本科生合作、指导博士后、从代表性不足的群体中培养博士生以及帮助代表性不足的群体从事研究方面的良好记录。

HeilDMS-0806532 Four main directions are selected for advancement in thisproject:(a) Time-frequency independence conjectures and related topics;(b) Quantification of localization for nonabelian coherent state frames;(c) Applications to affine pseudodifferential operators (broadband time-varying filters);(d) Multiwavelet development of the shearlet transform and construction of compactly supported shearlets.Each of these interrelating topics contains deep mathematicalquestions directly related to fundamental issues in redundantrepresentation and transformation of signals. Each topic reachesinto a broad range of areas in mathematical analysis, andprogress requires a synthesis of different ideas from bothabstract analysis and applied harmonic analysis, includingpseudodifferential operator theory, noncommutative Banachalgebras, and new forms of multiresolution analysis suitable fortwo-dimensional and higher signal processing. The problems addressed in this project concern fundamentalquestions regarding the basic mathematical tools that underliemodern signal processing, including such applications as wirelesscommunications, radar, seismic sensing, and image processing. Progress on these topics yields applicable mathematics withpotential impact for real-world scientific and engineeringprograms. All aspects of this work are intimately tied witheducational endeavors, both at the graduate and undergraduatelevels. Students of the investigator cooperate directly in thisresearch, receiving a breadth and depth of knowledge andexperience that will foster a continued commitment tocross-disciplinary interactions among mathematics, physics, andengineering. Topics related to this project and progress on theproblems of the project are incorporated into courses andseminars directed by the investigator. The investigatorcontinues his strong record of producing graduate students,working with undergraduates, mentoring postdocs, producing Ph.D.students from underrepresented groups, and encouragingunderrepresented groups to pursue research.