Topics in Banach Space Theory

巴拿赫空间理论主题

• 批准号: 0701552
• 负责人: Stephen Dilworth
• 金额: $ 11.52万
• 依托单位: University South Carolina Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701552&HistoricalAwards=false
• 关键词: Topics; Banach; Space; Theory

This project will investigate some problems in Banach space theory which are motivated in part by applications to signals processing and data compression. Elements of a given Banach space can be approximated by linear combinations of elements drawn from a Schauder basis or from a redundant system such as a frame or a dictionary. The approximants are typically selected by a greedy algorithm such as the Dual Greedy, X-Greedy , or Thresholding Greedy Algorithms. The project will study the convergence of such algorithms in the norm or the weak topology of the Banach space. An important open problem in this area is to show that the X-Greedy Algorithm converges in Lebesgue spaces. Convergence results for greedy algorithms are connected to geometrical properties of the underlying Banach space such as uniform smoothness or the Kadets-Klee property. They are also connected to ``partial unconditionality'' properties of the underlying system such as quasi-greediness. An important related problem in Banach space theory which will be studied is to show that certain absolute constants (the Elton constants) which arise naturally in the study of partial unconditionality are uniformly bounded. A second set of problems in this project concerns coefficient quantization in Banach spaces. The goal is to replace arbitrary real coefficients (with respect to some underlying system) by coefficients that are selected from a finite ``alphabet'' using some algorithmic procedure. Building on recent results in the case of a Schauder basis, existence of systems with desirable quantization properties will be connected to the geometry of the underlying Banach space. A Banach space is a collection of ``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 everyday notion of distance between points in the three-dimensional Banach space which we inhabit. There is a wide range of Banach spaces of importance in both pure and applied mathematics which can be distinguished from each other by ``geometrical'' properties such as ``smoothness'' or ``convexity''. Mathematicians have found that Banach spaces provide the appropriate framework in which to formulate major areas of mathematics such as Harmonic Analysis, Partial Differential Equations, and Functional Analysis. Banach spaces are also used by scientists and engineers to model problems in applied areas such as fluid mechanics, signals processing, image compression, and the pricing of financial derivatives. An individual vector belonging to a given Banach space is usually identified by an infinite string of numbers called ``coefficients'' . The problem of data compression is to select the most significant coefficients and to discard the rest. The problem of analog to digital conversion is to ``quantize'' the selected coefficients by binary numbers in such a way that the quantized vector is a good approximation to the ``target'' vector. The geometry of the underlying Banach space will play an important part in the development of effective algorithms for implementing such procedures. In this project we will investigate these and other problems concerning Banach spaces and their geometrical properties.

该项目将研究巴纳赫空间理论中的一些问题，这些问题部分是由信号处理和数据压缩的应用引起的。给定 Banach 空间的元素可以通过从 Schauder 基或冗余系统（例如框架或字典）提取的元素的线性组合来近似。近似值通常由贪婪算法选择，例如对偶贪婪算法、X-贪婪算法或阈值贪婪算法。该项目将研究此类算法在巴纳赫空间的范数或弱拓扑中的收敛性。 该领域的一个重要的开放问题是证明 X-贪婪算法在勒贝格空间中收敛。贪婪算法的收敛结果与基础 Banach 空间的几何属性（例如均匀平滑度或 Kadets-Klee 属性）相关。它们还与底层系统的“部分无条件”属性（例如准贪婪）相关。将要研究的巴纳赫空间理论中的一个重要相关问题是证明在部分无条件性研究中自然出现的某些绝对常数（埃尔顿常数）是一致有界的。该项目中的第二组问题涉及巴纳赫空间中的系数量化。目标是用使用某种算法程序从有限“字母表”中选择的系数来替换任意实数系数（相对于某些底层系统）。 基于 Schauder 基的最新结果，具有理想量化特性的系统的存在将与基础 Banach 空间的几何结构相关联。 Banach 空间是“向量”的集合，它们可以相加或乘以数字以形成其他向量。向量之间有一个“距离”的概念，类似于我们所居住的三维巴拿赫空间中点之间距离的日常概念。在纯数学和应用数学中存在着广泛的重要的巴纳赫空间，它们可以通过“几何”属性（例如“平滑性”或“凸性”）来区分。数学家发现巴拿赫空间提供了适当的框架来制定数学的主要领域，例如调和分析、偏微分方程和泛函分析。科学家和工程师还使用巴拿赫空间对流体力学、信号处理、图像压缩和金融衍生品定价等应用领域的问题进行建模。属于给定 Banach 空间的单个向量通常由称为“系数”的无限数字串来标识。数据压缩的问题是选择最重要的系数并丢弃其余的。模数转换的问题是通过二进制数“量化”选定的系数，使得量化的向量非常接近“目标”向量。 基础 Banach 空间的几何形状将在开发实现此类过程的有效算法中发挥重要作用。在这个项目中，我们将研究有关巴纳赫空间及其几何性质的这些问题和其他问题。