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

KrishtalDMS-0908239 该项目致力于创建，开发和应用框架，融合框架和模糊融合框架的矩阵式表示的统一理论;以及伪微分算子的容差理论和正则因子分解。 这些主题涉及基础和应用分析的各种领域，包括框架理论，伪微分算子理论，Banach模和局部紧阿贝尔群的表示，非标准多分辨率分析和信号处理。 在这个项目中研究的问题解决了有趣的和重要的问题，关于现代信号处理所基于的某些数学工具。 预计该项目还将在更基本的分析中提出新的问题，从而促进各领域的相互促进。 研究人员和他的同事们开发了一个理论框架，旨在解决现代世界中大量数据所带来的各种问题。 潜在的应用包括图像处理、无线通信等。特别注意质子计算机断层扫描的应用。 拟议活动的更广泛影响包括研究和教育的一体化、知识的传播以及国内和国际跨学科合作的加强。

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.