| Date: Friday, April 15, 2011
Title: The Gibbs and Runge Phenomena in Approximation Using Samples on a Uniform Grid: Remedies, Limits and Open Questions
Abstract: Gibbs Phenomenon is the non-uniform converge of a Fourier series or trigonometric interpolation to a function which is nonperiodic or otherwise has singularities on the approximation interval. The Runge Phenomenon is the divergence of polynomial interpolation on a uniform grid due to singularities of the target function in the complex plane. We describe a plethora single-interval, three-interval and multidomain strategies for mitigating the Gibbs and Runge Phenomena and explain their connections. Filtering, series acceleration, windows, Tikhonov Regularization and Fourier and radial basis function Extension are part of the toolbox. We discuss the limits, both intrinsic and floating point precision, on these algorithms, and end with numerous unresolved questions.
Speaker: John Boyd
Institution: University of Michigan, AOSS
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