Applications of Functional Analysis

泛函分析的应用

• 批准号: 9706108
• 负责人: Peter Casazza
• 金额: $ 5万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706108&HistoricalAwards=false
• 关键词: Applications; Functional; Analysis

9706108 Casazza The research project will use techniques from Functional Analysis to developed a local theory for wavelets and frames. The idea is to find the least number of elements from one of these sets to give a good approximation to problems in these areas. This will allow finite dimensional techniques to be used on infinite dimensional problems as well as giving quick approximations to solutions which are more accessible to the computer. We use, and extend, deep results from the local theory of Banach spaces to stronger results in Hilbert spaces to carry out the goals of the project. The project also involves work on the L(1)-Problem: Is every complemented subspace of L(1) isomorphic to L(1) or l(1)? A new approach to the problem relies on shifting the problem to an equivalent problem on a more computable space than L(1). This setting may allow a solution to the problem. The main goal is to reduce calculations of wavelets and frames to the least number of vectors possible. First, this will allow infinite dimensional calculations to be reduced to the finite dimensional setting. Second, in the finite dimensional setting, it may reduce the number of necessary calculations to something closer to the capacity of the computer. This could significantly reduce the length of time for calculations and for transmitting data. These methods are some of the most applied techniques around today. They are used in imaging problems from everything from satellite imaging to x-rays. If one could significantly reduce the calculations necessary for such computations, it should have broad application to all of these important imaging problems.

9706108 卡萨扎 该研究项目将使用功能分析的技术 发展了小波和框架的局部理论。 这个想法是从这些集合中找到最少数量的元素， 对这些领域的问题给出了很好的近似。 这将允许有限维技术被用于无限维的问题，以及快速近似的解决方案，更容易进入计算机。 我们使用，并扩展，从局部理论的Banach空间，更强的结果在希尔伯特空间进行项目的目标。 该项目还涉及工作 关于L（1）-问题：是L（1）的每个补子空间吗 同构于L（1）或l（1）？解决问题的新方法 依赖于将问题转化为一个 比L（1）更大的可计算空间。 此设置可能 允许解决问题。 主要目标是减少小波和框架的计算 到尽可能少的载体。 首先，这将使 无限维的计算可以简化为有限维的计算。 尺寸设置。 第二，在有限维环境中， 它可以减少必要的计算次数， 更接近计算机的容量。 这可以大大 缩短计算和传输时间 数据 这些方法是一些最实用的技术， 今天 它们用于成像问题， 卫星成像到X射线。 如果能大幅降低 对于这样的计算所需的计算，它应该有广泛的 应用于所有这些重要的成像问题。