Applications of Functional Analysis

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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.