Time-Frequency and Applied Harmonic Analysis

在这个项目中选择了四个主要方向来推进：(A)时频无关猜想和相关主题；(B)非阿贝尔相干态框架的局部化的量子化；(C)仿射拟微分算子(宽带时变滤波器)的应用；(D)剪切波变换的多小波发展和紧支剪切器的构造。这些相互关联的主题中的每一个都包含与信号的冗余、表示和变换中的基本问题直接相关的深层数学问题。每个主题都涉及到数学分析的广泛领域，而要取得进展，需要综合抽象分析和应用调和分析的不同思想，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新形式的多分辨分析。本项目所涉及的问题涉及现代信号处理的基本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。在这些主题上的进展产生了适用的数学，对现实世界的科学和工程项目具有潜在的影响。这项工作的所有方面都与教育努力密切相关，无论是在研究生还是本科生。研究人员的学生在这项研究中直接合作，获得了广度和深度的知识和经验，这将促进对数学、物理和工程学之间跨学科互动的持续承诺。与该项目有关的主题和该项目问题的进展被纳入由调查员指导的课程和研讨会。这位研究人员延续了他在培养研究生、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鼓励代表性不足的群体从事研究方面的强劲记录。

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.