Mathematical Sciences: Research Group in Analysis

摘要罗奇伯格、克兰茨、维克豪泽、韦斯、塔伊夫顿、麦卡锡 主要研究者建议成立一个分析研究小组。他们建议调查广泛的领域在谐波分析，算子理论，函数理论和微分方程。 具体的研究领域包括伯格曼和哈代空间的分析以及作用于它们的算子，复域几何与相关函数理论和算子理论之间的关系， 小波理论和相关的结构以及这些工具和思想在理论和计算问题中的应用，以及树的调和分析和探索树分析与连续调和分析之间的类比。 主要研究人员提出谐波分析（广义）的研究计划。 调和分析的基本统一观点之一是发展系统的技术，将复杂的对象分解成简单的，易于理解的，以简单的方式放在一起的碎片。 在许多情况下，这些最初是为了理解理论数学结构而开发的相同技术，也可以是 有效地适应于数据和图像的分析和处理。 最近的一个重要例子是新波形技术的发展，它为信号处理提供了重要的新工具。 详细的研究建议包括理论和应用组件。 后者集中在扩大适用范围的新的自适应波形技术，如小波。这些技术最初是为理论数学而开发的， 为信号处理提供了重要的新工具。 看来，这些想法有很大的潜力，进一步发展为计算和数据分析的工具。 这一发展是拟议研究的主题之一。

Abstract Rochberg, Krantz, Wickerhauser, Weiss, Taibleson, McCarthy The principal investigators propose to form a Research Group in Analysis. They propose to investigate a broad range of areas in harmonic analysis, operator theory, function theory, and differential equations. Specific areas of research include analysis of Bergman and Hardy spaces and the operators which act on them, the relationship between the geometry of complex domains and the associated function theory and operator theory, the theory of wavelets and related constructions and the applications of those tools and ideas in theoretical and computational problems, and harmonic analysis on trees and the exploration of the analogies between analysis on trees and continuous harmonic analysis. The principal investigators propose research programs in harmonic analysis (in the broad sense). One of the basic unifying viewpoints in harmonic analysis is the development of systematic techniques to decompose complicated objects into simple, well understood, pieces put together in a straightforward way. In many cases these same techniques, which were first developed to understand theoretical mathematical constructs, can also be adapted productively to the analysis and manipulation of data and images. An important recent example of this is the development of new waveform technologies which have provided important new tools in signal processing. The detailed research proposed includes both theoretical and applied components. The latter center on expanding the range of applicability of the new adapted waveforms techniques such as wavelets. Those techniques, originally developed for use in theoretical mathematics, have already provided important new tools in signal processing. It appears that they these ideas have great potential for further development as tools for computation and for data analysis. That development is one of the themes of the proposed research.