Matrix-like Representations in Time-Frequency and Applied Harmonic Analysis

KrishtalDMS-0908239该项目致力于框架、融合框架和模糊融合框架的类矩阵表示的统一理论的创建、发展和应用；以及伪微分算子的因果理论和正则因式分解。这些主题涉及基础和应用分析的许多领域，包括框架理论、伪微分算子理论、Banach模和局部紧Abelian群的表示、非标准多分辨分析和信号处理。这个项目中研究的问题解决了与现代信号处理所基于的某些数学工具有关的有趣和重要的问题。预计该项目还将在更基本的分析中引发新的问题，从而促进田间的交叉施肥。这位研究人员和他的同事开发了一个数学框架，旨在解决现代世界数据泛滥带来的各种问题。潜在的应用包括图像处理、无线通信等。特别关注质子计算机层析成像的一个应用。拟议活动的更广泛影响包括整合研究和教育、传播知识以及加强国内和国际的跨学科合作。

KrishtalDMS-0908239 The project is dedicated to creation, development, andapplication of a unified theory of matrix-like representationsfor frames, fusion frames, and fuzzy fusion frames; and causalitytheory and regular factorizations of pseudodifferentialoperators. These topics touch upon a large variety of areas offundamental and applied analysis, including frame theory,pseudodifferential operator theory, Banach modules andrepresentations of locally compact Abelian groups, non-standardmultiresolution analysis, and signal processing. The problemsstudied in this project address interesting and importantquestions regarding certain mathematical tools on which modernsignal processing is based. The project is also expected to leadto new questions in more fundamental analysis, thus promotingcross-fertilization of the fields. The investigator and his colleagues develop a mathematicalframework aimed to address various problems posed by the delugeof data in the modern world. Potential applications includeimage processing, wireless communications, and others. Particular attention is given to an application for protoncomputed tomography. Broader impact of the proposed activitiesincludes integration of research and education, dissemination ofknowledge, and enhancement of interdisciplinary collaboration,both domestic and international.