9401208 Guo This RUI award supports mathematical research on problems in harmonic analysis related to spectral synthesis, which roughly states that smooth, decaying functions vanishing off a set approximate Fourier transforms vanishing on the same set. This project is mainly concerned with spectral synthesis on manifolds, the restriction of the Fourier transform and partial differential equations with constant coefficients. Specifically, the project concerns distributions supported by hypersurfaces in Euclidean space and how the curvature of the manifold is related to decay properties of the Fourier transform of the distribution. The work draws on recent results giving asymptotic estimates for weak spectral synthesis. Since there are examples where weak spectral synthesis fails for general manifolds, it is necessary to investigate the proper conditions, such as constant nullity, which will provide positive answers. Harmonic analysis concerns the decomposition and reconstruction of functions through representations in terms of simpler components. One of the fundamental tools of harmonic analysis is the Fourier transform which plays equally central role in the study of partial differential equations where the two fields come together naturally. Analysis of the transform, restricted to manifolds, is a relatively open field in which general results are usually hard to achieve.
