Applications of Functional Analysis

这个研究项目将使用泛函分析的技术来发展小波和框架的局部理论。我们的想法是从这些集合中找出最少数量的元素，从而对这些区域的问题给出一个很好的近似。这将允许将有限维技术用于无限维问题，并给出更易于计算机访问的快速近似解。我们利用Banach空间的局部理论的深度结果，并将其推广到Hilbert空间的更强结果，以实现项目的目标。该项目还涉及到L(1)问题的工作：L(1)的每个补子空间是否同构于L(1)或L(1) ？一种新的方法依赖于将问题转移到比L(1)更可计算的空间上的等效问题。此设置可能允许解决该问题。主要目标是将小波和帧的计算减少到尽可能少的向量。首先，这将允许无限维度的计算减少到有限维度的设置。其次，在有限维度的情况下，它可以减少必要的计算次数，使其更接近计算机的能力。这可以大大减少计算和传输数据的时间长度。这些方法是当今应用最广泛的一些技术。它们被用于从卫星成像到x射线的各种成像问题。如果能显著减少这种计算所需的计算量，它将广泛应用于所有这些重要的成像问题。

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.