Time-Frequency and Applied Harmonic Analysis

本课题选择了四个主要的研究方向：(a)时频无关猜想及相关课题；(b)非abel相干状态帧的局部化量化；(c)仿射伪微分算子（宽带时变滤波器）的应用；(d) shearlet变换的多小波发展和紧支shearlet的构造。这些相互关联的主题都包含与信号冗余表示和变换的基本问题直接相关的深层数学问题。每个主题都涉及数学分析的广泛领域，并且需要从抽象分析和应用谐波分析中综合不同的思想，包括伪微分算子理论，非交换Banachalgebras，以及适用于二维和更高信号处理的新形式的多分辨率分析。本项目解决的问题涉及现代信号处理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程序具有潜在影响的应用数学。这项工作的所有方面都与研究生和本科生的教育努力密切相关。研究者的学生直接在这项研究中合作，获得广度和深度的知识和经验，这将促进数学，物理和工程之间跨学科互动的持续承诺。与该项目相关的主题和项目问题的进展被纳入由研究者指导的课程和研讨会。这位研究者在培养研究生、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鼓励代表性不足的群体从事研究方面保持着良好的记录。

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.