Topics in the geometry of Banach spaces

Banach 空间几何主题

• 批准号: 1001929
• 负责人: Daniel Freeman
• 金额: $ 7.64万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001929&HistoricalAwards=false
• 关键词: Topics; geometry; Banach; spaces

This project specifically addresses problems in the geometry of Banach spaces with a focus on the analysis of coordinate systems such as bases and frames. Coordinate systems are widely used in both application and theory, strongly connecting these problems to other areas of mathematics such as approximation theory, descriptive set theory, and differential topology. For example, new Banach spaces are often constructed by explicitly building a basis for the space. In this spirit, this project will address the construction of new Banach spaces with the property that every bounded operator on the space is a multiple of the identity plus a compact operator. Moreover, the project will also study the greedy approximation properties of coordinate systems in Banach spaces. Greedy approximation is based on the idea of always taking the "biggest piece" in each step of an iterative algorithm. This project will consider the existence of greedy bases and the convergence of greedy algorithms in particular Banach spaces. Beyond greedy approximation, this project intends to extend the descriptive set theory approach to bases, which has given remarkable insight into the structural theory of Banach spaces, to that of frames. Furthermore, this project will work on adapting the techniques and structure of Hilbert and Banach frames to the continuously varying setting of vector bundles.The structural attributes of Banach spaces and Hilbert spaces make them ideal settings for analyzing many problems in mathematics and engineering. A common example is encoding and transmitting signals. Bases in a Hilbert space or Banach space give a unique representation for the vectors in the space while the representation given by a frame is redundant. Signal encoding and transmission is often accomplished by sending coefficients with respect to some basis. This strategy, however, is not robust in the face of error, as any loss or corruption of basis coefficients results in the loss of entire dimensions of the signal. This is where frames come in as their redundancy distributes error loss over the whole space instead of concentrating it in isolated dimensions. Frames now play an important role in signal processing, and the study of their geometry in both Hilbert and Banach spaces is a growing area of research. Additionally, sometimes it is important to consider not just a single vector space, but some related collection of spaces. For example, the tangent bundle of a surface is the collection of tangent planes to the surface. In this case we want a basis for the tangent space at each point which moves smoothly over the surface. It is impossible to find such a basis for many surfaces. On the contrary, it is always possible to find a redundantframe for the tangent space which moves smoothly. Given this, it is naturally of interest to study such frames.

该项目专门解决了Banach空间的几何形状中的问题，重点是对坐标系（例如底座和框架）的分析。 坐标系在应用和理论中广泛使用，将这些问题与其他数学领域（例如近似理论，描述性集理论和差异拓扑结构）进行了强烈联系。 例如，新的Banach空间通常是通过明确为空间建立基础来构建的。本着这种精神，该项目将解决新的Banach空间的建设，并使用该物业的财产来解决该空间上每个有界操作员的倍数以及紧凑的操作员的倍数。此外，该项目还将研究Banach空间中坐标系的贪婪近似特性。贪婪的近似是基于在迭代算法的每个步骤中始终采取“最大作品”的想法。该项目将考虑贪婪基地的存在以及贪婪算法在特别是Banach空间中的收敛性。 除了贪婪的近似之外，该项目旨在将描述性的理论方法扩展到基础上，该方法对Banach空间的结构理论有了极大的了解。此外，该项目将致力于将希尔伯特和巴纳克框架的技术和结构调整为矢量捆绑包的不断变化的设置。BanachSpace和Hilbert Space的结构属性使它们成为分析数学和工程学中许多问题的理想设置。一个常见的示例是编码和传输信号。希尔伯特（Hilbert）空间或巴拉奇（Banach）空间中的基地为空间中的向量提供了独特的表示，而框架给出的表示为冗余。 信号编码和传输通常是通过发送系数相对于某些基础来完成的。但是，面对错误，这种策略并不强大，因为基础系数的任何损失或损坏都会导致信号的整个维度损失。这是框架进入的地方，因为它们的冗余在整个空间上分布了错误损失，而不是将其集中在孤立的维度上。现在，框架在信号处理中起着重要的作用，对希尔伯特和巴纳克空间的几何形状的研究是一个越来越多的研究领域。此外，有时不仅要考虑一个向量空间，而且要考虑一些相关的空间集合很重要。例如，表面的切线束是向表面的切线的集合。 在这种情况下，我们需要在每个点处的切线空间的基础，该空间在表面上平稳移动。为许多表面找到这样的基础是不可能的。相反，始终有可能为切线平稳的切线空间找到一个冗余帧。 鉴于此，研究此类框架自然是感兴趣的。