Mathematical Sciences: Real Variable Techniques in the Approximation of Functions and Boundary Value Problems in Nonsmooth Domains

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ABSTRACT Proposal: DMS-962325 PI: Torres Torres will continue with his work in harmonic analysis and partial differential equations. This work combines wavelet decompositions, singular integrals, and related real variable techniques. He plans to study functions that can be characterized by their values on a discrete set of points. A scale of discrete function spaces introduced by Torres in his previous work quantifies the oscillatory properties of these functions. He will now use these spaces to measure the errors in various linear and nonlinear approximations of band-limited signals. He intends to provide optimal estimates that cannot be attained with more classical techniques which only take into account the size of the functions. Other technical aspects of this part of the research relate to problems about operator theory and Besov spaces. In the second part of his program, Torres will continue his work on transmission problems in nonsmooth domains. For example, he will consider problems arising in electromagnetism. He will use singular integrals and layer potentials techniques. He also wants to combine his work with recent numerical analysis techniques for the study of large systems of linear equations. Band-limited signals are functions obtained as superposition of periodic waves whose frequencies remain bounded. Most signals in applications are band-limited and they can be recovered from the knowledge of their values on a discrete set of instants in time. This is very important for practical purposes because, often, a set of measured samples of a signal is all that is available from an experiment or a detection procedure. Fourier and Harmonic Analysis techniques codify, transform, and interpret the information and properties of a function or signal in a quantified way. In particular, the discrete character of wavelets and the techniques to be used make them amenable to numerical applications. Error estimates in approximations and the processes to minimize them trans late into results about compression of information. Other related problems are found in tomography and medical imaging. Transmission problems arise in physical situations in which an object interacts with the surrounding ambient space or when an object is composed of several different materials. These problems require the simultaneous solution, inside and outside of a region in space, of a systems of differential equations that are coupled by conditions on the boundary of the region. Effective tools for studying such problems in smooth domains are available, but it is very important to further develop the theory so it can adapted to domains that have corners or edges since they model realistic situation in the applied sciences.