Topics in Banach Space Theory

这个项目将研究Banach空间理论中的一些问题，这些问题的部分动机是应用于信号处理和数据压缩。给定Banach空间的元素可以由从Schauder基或从框架或字典等冗余系统中提取的元素的线性组合来近似。通常通过贪婪算法来选择近似值，例如双重贪婪、X-贪婪或阈值贪婪算法。该项目将研究这类算法在Banach空间的范数或弱拓扑下的收敛。在这一领域中一个重要的公开问题是证明X-贪婪算法在勒贝格空间中收敛。贪婪算法的收敛结果与Banach空间的几何性质有关，如一致光滑性或Kadets-Klee性质。它们还与基本系统的“部分无条件”性质有关，如准贪婪。Banach空间理论中一个重要的相关问题是证明在研究部分无条件时自然产生的某些绝对常数(Elton常数)是一致有界的。该项目中的第二组问题涉及Banach空间中的系数量化。目标是用使用某种算法程序从有限的“字母表”中选择的系数来替换任意的实数系数(相对于某些基本系统)。在Schauder基的最新结果的基础上，具有理想量子化性质的系统的存在将与基础Banach空间的几何相联系。Banach空间是“向量”的集合，这些“向量”可以相加或乘以数字以形成其他向量。有一个向量之间的“距离”的概念，类似于我们居住的三维Banach空间中点之间的距离的日常概念。在纯数学和应用数学中都有广泛的重要的Banach空间，它们可以通过“几何”性质，如“光滑性”或“凸性”相互区分。数学家们已经发现，Banach空间提供了适当的框架，在其中可以表达主要的数学领域，如调和分析、偏微分方程组和泛函分析。科学家和工程师还使用Banach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一个称为“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系数。模数转换的问题是用二进制数对所选系数进行“量化”，使量化后的向量与“目标”向量很好地接近。底层Banach空间的几何将在开发实现此类过程的有效算法方面发挥重要作用。在这个项目中，我们将研究这些和其他有关Banach空间及其几何性质的问题。

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.