Mathematics Colloquia and Seminars

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the manipulation of piecewise smooth data. Examples include images, which are a collection of objects distinguished by edges, or the solutions to time dependent problems, where initially smooth data may spontaneously develop discontinuities. The majority of modern approximation methods have a fixed, polynomial order of accuracy, for example splines and the essentially non-oscillatory polynomial approximations. Although global projections yield exponentially close approximations for smooth functions, a single discontinuity introduces O(1) spurious oscillations, Gibbs' Phenomena, and reduces the high order convergence rate to first order. I will present Semi-global methods which, by adapting to the discontinuity locations, recover the exponential convergence rate, as well as remove the spurious oscillations. Additionally, efficient implementations are achieved through construction in the frequency space(filters), and the physical space(mollifiers). Applications to time dependent problems will be shown. Time permitting, I will also give a brief overview of ongoing research in multi-atom Gabor decompositions for the more efficient time-frequency extraction of local tonal content in a signal, and the recovery of a band-limited signal from its oversampled values. These research projects were conducted in collaboration with Eitan Tadmor, Thomas Strohmer, and David Gottlieb.