Banach Spaces and Applications

Banach 空间和应用

• 批准号: 1361461
• 负责人: Stephen Dilworth
• 金额: $ 14.72万
• 依托单位: University of South Carolina at Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361461&HistoricalAwards=false
• 关键词: Banach; Spaces; Applications

A Banach space is a collection of objects called vectors which can be added together or multiplied by numbers to form other vectors. There is a concept of distance between vectors which is analogous to the familiar notion of distance between the points in the three-dimensional world which we inhabit. Mathematicians have found that Banach spaces provide the correct framework in which to formulate major areas of mathematics such as Functional Analysis and Partial Differential Equations. Banach spaces are also used by scientists and engineers to model problems in applied areas such as fluid mechanics, signals processing, and finance. There are an infinite variety of Banach spaces which can be distinguished from each other by geometrical properties such as smoothness and convexity. An individual vector belonging to a Banach space is identified by an infinite string of numbers called coefficients. An important problem in data compression is to find a procedure, sometimes called a greedy algorithm, for selecting the most significant coefficients so that the resulting finite string vector is a short distance from, and hence a good approximation to, the original vector. This project will investigate fundamental problems in Banach space theory and applications to other areas. The methods employed will be those of Functional Analysis together with new insights specific to each particular problem. These problems include the Banach Rotation Problem which asks whether Hilbert space is the only separable infinite-dimensional Banach space with a transitive isometry group. The new notion of asymptotic midpoint convexity will also be investigated. An open question to be solved is whether the isomorphic version of this property is equivalent to the known concept of asymptotic uniform convexity. Another problem to be solved in the area of nonlinear Banach space theory is whether $p$-convexity is preserved under uniform quotient mappings. Applications to other areas to be investigated include open questions on the convergence of greedy algorithms in Banach spaces, including the important case of Lebesgue spaces, and related open questions on unconditionality and greedy convergence, including the open problems of the existence of quasi-greedy sequences in arbitrary Banach spaces and of the boundedness of the Elton constants. Other applications include improved explicit constructions of matrices with the Restricted Isometry Property, and coefficient quantization properties for bases and redundant systems.

一个Banach空间是一个被称为向量的对象的集合，这些对象可以被加在一起或乘以数字以形成其他向量。向量之间有一个距离的概念，它类似于我们所熟悉的三维世界中点之间距离的概念。数学家们发现，Banach空间提供了正确的框架，在其中制定数学的主要领域，如功能分析和偏微分方程。Banach空间也被科学家和工程师用于模拟应用领域的问题，如流体力学，信号处理和金融。有无限多的Banach空间，可以通过几何性质如光滑性和凸性来相互区分。属于Banach空间的单个向量由称为系数的无限串数字标识。数据压缩中的一个重要问题是找到一个过程，有时称为贪婪算法，用于选择最重要的系数，使得所得的有限字符串向量与原始向量的距离很短，因此是原始向量的良好近似。本项目将研究Banach空间理论的基本问题及其在其他领域的应用。所采用的方法将是那些功能分析与新的见解，具体到每个特定的问题。这些问题包括巴拿赫旋转问题，其中要求希尔伯特空间是否是唯一可分的无限维巴拿赫空间与传递等距群。渐近中点凸性的新概念也将被研究。一个有待解决的公开问题是这个性质的同构版本是否等价于已知的渐近一致凸性概念。在非线性Banach空间理论中的另一个问题是在一致商映射下是否保持p-凸性。应用到其他领域的调查包括开放问题的收敛性贪婪算法在Banach空间，包括重要的情况下勒贝格空间，和相关的开放问题的无条件性和贪婪的收敛性，包括开放问题的存在准贪婪序列在任意Banach空间和有界性的埃尔顿常数。其他应用包括改进的显式构造矩阵的限制等距属性，和系数量化属性的基地和冗余系统。